Midterm 2 Study Guide
•
•
•
•
•
•
You are allowed to bring a single 8.5 x 11-inch sheet of notes (both sides.)
Please use a pencil and eraser, not a pen.
You may use a non-cell phone, non-graphing calculator that does not have matrix functionality.
Since food aromas and crinkling wrappers are distracting, please refrain from eating during the test.
You may not leave the room or have your phone out during the test.
Bring a photo ID if you don’t attend class regularly.
The exam will cover sections 1.8, 1.9, 2.1, 2.2, 2.3, 2.8, 2.9
You should be able to:
1.
For linear transformations: T(x) = Ax
a. Find the image of a vector under a transformation 𝑇.
b. Find the standard matrix, A for a linear transformation 𝑇: ℝ! → ℝ" .
i. If given T(e1), T(e2), etc.
ii. If given a “formula” for T(u) - see practice exercises
iii. If given written instructions, i.e rotated by a certain angle, reflected across an access, etc.
c. Determine if a given vector is in the range of a transformation., and if so, find a vector that maps to it.
d. Determine the domain and codomain of a linear transformation T(x) = Ax, given A
e. If given size of domain and codomain, determine size of A
f. Determine if transformation is one-to-one or onto
2. Perform matrix arithmetic (such as adding/subtracting matrices, multiplying a matrix by a scalar, multiplying two
matrices.)
3. Invertible Matrices
a. Determine if a matrix is invertible.
b. Find the inverse (if it exists) of a small matrix by hand.
c. Use an inverse matrix to solve a linear system of equations.
d. Understand the statements in the Invertible Matrix Theorem and be able to apply them.
4. For a set of vectors in Rn
a. Determine if it is a basis for all of Rn
b. For the subspace equal to the span of the set of vectors
i. Find a basis
ii. Determine the dimension of the subspace
5. For NulA
a. Determine if a vector is in NulA
b. Find vectors in NulA
c. Find a basis for NulA
d. Determine dim(NulA)
6. For Col A
a. Determine if a vector is in ColA
b. Find vectors iin ColA
c. Find a basis for ColA
d. Determine dim(ColA)
7. For a given matrix A, determine its rank
8. Understand the relationships between the size of matrix A, rank(A), dim Nul(A), and use these relationships to solve
problems.
9. Given a basis for a subspace,
a. if given a vector x find xB, the coordinate matrix for the basis B
b. if given a vector xB find the vector x
10. Answer true/false questions and provide reasons for your answers.
0
