MAT188 Linear Algebra -- My Learning Standards Tracker
University of Toronto
Fall 2026
Faculty of Applied Science & Engineering
Name:
Track your own progress on every learning standard. The magenta code in brackets is the standard’s
key — the same code you will see on the readings, tutorials and quizzes.
Practise all of them. Only some of them are assessed. Every standard on this list is
something you should be able to do by the end of the course, and every one of them can appear in
an ordinary question on a midterm or the final. Only a selection of them comes up in a tutorials
or a Numerical Mastery practices. Those are the ones you get to practice with the help of your
TAs and that can carry a mastery record. The rest are yours to practice on your own and bring
questions to drop-in hours.
How to use each column:
ˆ Practice to master — tick the box once you can do this standard on your own, without notes
and without help. This is your own judgement, and it is the column that matters most. An
empty box means more practice needed.
ˆ Showed mastery in — tick where you were assessed and shown to have mastered it: in a
tutorial SBG-TUT, on a SBG Numerical Computation test (Gateway), or in the last-chance
section of a Test (a midterm or the final). Many standards are never assessed; leaving this
blank is normal and is not a gap.
ˆ Notes — anything to remind yourself (a tricky case, a mistake to avoid, a page to reread).
Learning standard (I can. . . )
I Practiced
to master
I showed
mastery in
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Notes
Vectors, lines and planes
I can describe sets using set builder notation.
[ch0-WRIT-sets]
I can perform algebraic arithmetic on vectors
in a Euclidean vector space Rn (e.g., addition,
scalar multiplication, norm, and dot product).
[ch0-COM-vecarith]
I can represent vectors arithmetic geometrically for vectors in R2 and R3 .
[ch0-VG-vecarith]
I can decide whether given vectors are parallel.
[ch0-COM-par]
I can decide whether given vectors are perpendicular.
[ch0-COM-per]
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Learning standard (I can. . . )
Practice to
master
Showed
mastery in
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I can compute the angle between any two vectors.
Notes
[ch0-COM-angle]
I can represent a line in R2 and R3 in vector
form.
[ch0-CON-vecline]
I can represent a line in R2 and R3 in parametric form.
[ch0-CON-parline]
Given an algebraic description of a line in R2
or R3 , I can visualize the line.
[ch0-VG-line]
I can represent a plane in R3 given a normal
vector and a point.
[ch0-CON-plane]
Given an algebraic description of a plane, I
can visualize the plane.
[ch0-VG-plane]
Linear systems and matrices
I can perform row reduction on any matrix
and reduce it to REF and RREF.
[ch1-COM-rref]
I can determine when a matrix is in REF or
RREF form.
[ch1-CON-rref]
Given an augmented matrix of a linear system, I can find the general solution to the
system.
[ch1-COM-augset]
Given the general solution to a system of linear equations, I can find particular solutions.
[ch1-COM-pslinsys]
Given the REF of an augmented matrix, I can
determine the type of solution set the system
has (no solution, a unique solution, infinitely
many solutions).
[ch1-CON-augsoltype]
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Learning standard (I can. . . )
Practice to
master
Showed
mastery in
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I can identify the solution type of a system
based on information about the rank of its
augmented and coefficient matrices.
Notes
[ch1-CON-ranksoltype]
Given a system of linear equations in R2 or
R3 , I can visualize the system and its general
solution.
[ch1-VG-gslinsys]
Given the matrix form A⃗x = ⃗b where ⃗b is a
vector in R2 or R3 , I can visualize a particular
solution for the system by writing ⃗b as a linear
combination of the columns of A.
[ch1-VG-pslinsys]
I can visualize the linear combination of vectors in R2 and R3 .
[ch1-VG-lincom]
I can interpret matrix-vector multiplication in
terms of a linear combination of the columns
of the matrix.
[ch1-CON-matlincom]
I can interpret matrix-vector multiplication
using dot products.
[ch1-CON-matdotprod]
I can add matrices of the same size, multiply a
matrix by a scalar and I can multiply a matrix
by a vector if the dimensions match up.
[ch1-COM-matarith]
I can describe the general solution of a system
parametrically and as a linear combination of
vectors using correct mathematical notation.
[ch1-WRIT-gslinsys]
I can describe a system in matrix-vector multiplication notation.
[ch1-WRIT-linsysnot]
Linear transformations, matrix algebra and inverses
Given a linear transformation in any form, I
can determine its domain and codomain.
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[ch2-CON-domcodom]
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Learning standard (I can. . . )
Practice to
master
Showed
mastery in
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Given a function from Rm to Rn I can apply
the definition of a linear transformation or the
Matrices theorem to decide whether it is a
linear map.
Notes
[ch2-CON-lintranscheck]
Given a linear transformation from Rm to Rn
I can compute its standard matrix by computing the output of the standard vectors.
[ch2-COM-standmat]
Given a linear transformation T in any form
I can find T (⃗x) for any vector ⃗x.
[ch2-COM-lintransoutput]
Given a linear transformation with domain
and codomain in R2 or R3 I can visualize the
effect of the transformation on the unit square
or the unit cube.
[ch2-VG-unitlintrans]
I can find (not memorize) the standard matrix
representation of the geometric transformations (rotation, reflection, projection, shear,
scaling) on R2 .
[ch2-CON-geotrans]
I can visualize the effect of the geometric linear transformations on R2 and R3 .
[ch2-VG-geotrans]
Given two linear transformations T and S, in
any form, I can compute the standard matrix
of T ◦ S or S ◦ T or show why they are not
defined.
[ch2-CON-lintranscomp]
I can visualize the effect of the composition of
two linear transformations on a given vector
in R2 and R3 .
[ch2-VG-comp]
Given matrices A and B, I can compute AB, if
defined, by applying A to columns of B using
the matrix-vector multiplication.
[ch2-COM-matvec]
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Learning standard (I can. . . )
Practice to
master
Showed
mastery in
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Given matrices A and B, I can compute AB,
if defined, by finding the ij-th entry of AB directly by computing the dot product between
the ith row of A and the jth column of B.
Notes
[ch2-COM-matdotprod]
I know the algebraic properties of matrix
arithmetic.
[ch2-COM-matarith]
Given a linear transformation T and its standard matrix A, I can perform row reduction
on A to decide whether T is invertible.
[ch2-COM-rrefinv]
Given a linear transformation T and its standard matrix A, if T is invertible, I can find
its inverse map T −1 , and the standard matrix
representation of T −1 , denoted by A−1 .
[ch2-COM-inv]
Given a linear transformation T : Rm → Rn I
can decide whether T is injective, surjective,
and/or invertible or bijective.
[ch2-CON-surinj]
Determinants
I can visualize the absolute value of the determinant of a 2 × 2 and determinant of a 3 × 3
matrix as a factor by which the corresponding
linear transformation changes the area of the
unit square or the volume of the unit cube,
respectively.
[ch6-VG-unit]
I can interpret the sign of the determinant of
a matrix geometrically.
[ch6-VG-detsign]
I can compute the determinant of an arbitrary
square matrix.
[ch6-COM-det]
I can use the fact that the determinant
is multilinear in rows and columns to simplify/compute the determinant of a matrix.
[ch6-CON-bilin]
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Learning standard (I can. . . )
Practice to
master
Showed
mastery in
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Given square matrices A and B, and a scalar
k, I can relate the det(AB), det(AT ), det(kA)
and det(A−1 ) to det A and det B.
Notes
[ch6-COM-detprop]
I can use determinant to determine whether
a given linear transformation is invertible.
[ch6-CON-invcheck]
Given a 2 × 2 matrix A, I can compute the
inverse of A using the matrix inverse formula.
[ch6-COM-inv]
I can use row reduction or multiplication by
elementary matrices to compute the determinant of a matrix.
[ch6-CON-rowred]
Subspaces, kernel and image, basis and coordinates
I can visualize all the subspaces of R, R2 , and
R3 .
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[ch3-VG-subspace]
Given a subset of Rn , I can verify whether it
is a subspace or not.
[ch3-CON-subspacecheck]
Given a set of vectors ⃗v1 , · · · , ⃗vn in R, R2 or
R3 I can visualize span(⃗v1 , · · · , ⃗vn ).
[ch3-VG-span]
Given a set of vectors ⃗v1 , · · · , ⃗vk in Rn I can
describe span(⃗v1 , · · · , ⃗vk ) and decide whether
any given vector is in it or not.
[ch3-CON-span]
I know the definition of the kernel and image
of a linear transformation T and can decide if
a given vector is in ker(T ) or im(T ), and can
find vectors in ker(T ) and im(T ).
[ch3-CON-imkerdef]
Given a linear transformation T , I can describe im(T ) in set builder notation.
[ch3-WRIT-im]
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Learning standard (I can. . . )
Practice to
master
Showed
mastery in
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Given a linear transformation T , I can describe the ker(T ) and im(T ) as a span of vectors.
Notes
[ch3-WRIT-ker]
Given a linear transformation T with domain
or codomain R, R2 , R3 I can visualize ker(T )
and im(T ).
[ch3-VG-imker]
Given a linear transformation T (⃗x) = A⃗x, I
can connect ker(T ) and im(T ) to the solutions
to A⃗x = ⃗b, for suitable choices of ⃗b.
[ch3-CON-gsker]
Given a linear transformation T , I can connect the notion of injective/surjective to
ker(T )/im(T ).
[ch3-CON-injkersurim]
I can determine if a given a set of vectors is
linearly independent.
[ch3-COM-linind]
Given a subspace, I can determine a basis.
[ch3-COM-bas]
Given a subspace, I can determine its dimension.
[ch3-COM-dim]
Given a set of vectors, I can reduce it to a
linearly independent set.
[ch3-COM-reduclinind]
Given a set of vectors, I can visualize if a vector is redundant.
[ch3-VG-redunvec]
Given a linear transformation T (⃗x) = A⃗x, I
can connect ker(T ) with the linear independence of columns of A and im(T ) with the
span of the columns of A.
[ch3-CON-kerlinind]
I can connect the dimension of the kernel of a
linear transformation to the dimension of its
image through Rank-Nullity Theorem.
[ch3-CON-kerrank]
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Learning standard (I can. . . )
Practice to
master
Showed
mastery in
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Given a subspace W of Rn described via a
basis, I can construct other bases for W .
Notes
[ch3-COM-bases]
I can take any vector ⃗v in any Rn , take a basis
B for Rn , and find the B coordinates of ⃗v in
Rn .
[ch3-COM-basvec]
Given any two bases B1 , B2 of Rn , I can construct a matrix (change of basis matrix from
B1 to B2 ) that take the B1 coordinates of ⃗v as
an input and compute the B2 coordinates of
⃗v as an output.
[ch3-COM-matbas]
Given a vector ⃗v ∈ Rn (n = 1, 2 or 3) and
bases B1 , B2 of Rn , I can visualize [⃗v ]B1 and
[⃗v ]B2 .
[ch3-VG-vecbas]
Given a linear transformation T : Rn → Rn ,
and a basis B for Rn , I can compute the matrix of T with respect to B and use it to find
output of vectors.
[ch3-COM-sbasmat]
I can connect the matrix representation of a
linear transformation T with respect to two
different bases, using a change of basis matrix.
[ch3-CON-cbasmat]
Given a geometric transformation T in R2 , I
can find a basis B (assuming it exists), such
that T with respect to B is diagonal.
[ch3-VG-geobasis]
Given a basis B such that a linear transformation T is diagonal, I can visualize T
[ch3-VG-diaglintrans]
Orthogonality, projection and least squares
Given a set of vectors, I can determine if they
are orthogonal, orthonormal, or neither.
[ch5-CON-orthotype]
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Learning standard (I can. . . )
Practice to
master
Showed
mastery in
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Given a subspace V , I can describe the orthogonal complement of V .
Notes
[ch5-COM-orthocomp]
Given a vector ⃗x in Rn and a subspace V ⊆
Rn , I can decompose ⃗x into two components,
one in V and one in the orthogonal complement of V .
[ch5-COM-vecdecomp]
Given a vector ⃗x in Rn and an orthonormal
basis U, I can find [⃗x]U , the coordinates of ⃗x
with respect to U without row reduction.
[ch5-COM-orthobas]
I can visualize the kernel and image of the
orthogonal projection onto a subspace V of
R2 , R3 .
[ch5-VG-kerim]
Given a basis B for a subspace V , I can apply the Gram-Schmidt process to transform B
into an orthonormal basis of V .
[ch5-COM-gram]
I can recognize if a linear transformation is
orthogonal.
[ch5-CON-ortho]
I can visualize the effect of an orthogonal linear transformation on the unit square/cube.
[ch5-VG-unitortho]
I understand computational properties of
transposes.
[ch5-COM-trans]
Given an inconsistent system A⃗x = ⃗b, I can
find ⃗x∗ such that A⃗x∗ is closest to ⃗b.
[ch5-CON-inconsys]
I can connect projections onto a subspace to
fitting a line to a set of data points.
[ch5-CON-projfit]
Eigenvalues and diagonalization
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Learning standard (I can. . . )
Practice to
master
Showed
mastery in
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Given a linear transformation T (⃗x) = A⃗x, I
can connect the diagonalizability of A to the
existence of an eigenbasis for T .
Notes
[ch7-CON-diaglintrans]
Given a geometric transformation in R2 , I
can visualize its eigenvectors and their corresponding eigenvalues.
[ch7-VG-geotrans]
Given a linear transformation T (⃗x), I can verify whether a vector ⃗v is an eigenvectors and if
so, determine the corresponding eigenvalues.
[ch7-COM-eigenvalue]
Given a linear transformation A⃗x, I can find
its eigenvalues and their algebraic multiplicity
using the characteristic polynomial.
[ch7-COM-algmult]
Given a characteristic polynomial of a 2 ×
2 matrix, I can determine determinant and
trace of A.
[ch7-COM-charpoly]
Given a linear transformation T (⃗x) and eigenvalue λ, I can find a basis for the eigenspace
corresponding to λ and its geometric multiplicity.
[ch7-COM-eigenbas]
Given a linear transformation in R2 , R3 , I can
visualize all the eigenspaces of T .
[ch7-VG-eigenspace]
Given a linear transformation T , I can state
and use a condition regarding the diagonalizability of T , in terms of the algebraic and
geometric multiplicities of its eigenvalues.
[ch7-CON-diagmult]
I can diagonalize a matrix A, if possible, by
finding a change of basis matrix C and a diagonal matrix D such that A = CDC −1 .
[ch7-CON-diagcmat]
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Learning standard (I can. . . )
Practice to
master
Showed
mastery in
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Given a diagonalizable A, I can predict long
term behavior of a dynamical system whose
transition matrix is A
[ch7-CON-dynsys]
Symmetric matrices
Given a symmetric matrix A, I can orthogonally diagonalize A (or orthogonally diagonalize the linear map whose standard matrix
is A).
[ch8-COM-diagsym]
Given a symmetric matrix A, I can visualize
the eigenspaces of A
[ch8-CON-specthm]
General skills
I can correctly use mathematical notation.
[chg-WRIT-matnot]
I can correctly and effectively communicate
ideas in mathematical language.
[chg-WRIT-matcom]
Given a statement I can decide whether it is
true or false, and correctly justify my decision.
[chg-CON-tf]
Given a description of a mathematical object
I can find an explicit example that satisfies
that definition.
[chg-CON-exp]
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Notes