MATH 115A – Linear Algebra
Theorem Sheet
If we’ve gone over a result in class and it’s not on here, then you can use it without citiation.
A good example is the cancellation theorem for a field or vector space. When you write −x,
you’re implicitly using the result that there is a unique vector y with x + y = 0. You don’t
have to cite this each time you write −x.
We assume that V is a vector space over a field F .
Theorem 1. Let S be a subset of V . Then Span(S) is a subspace of V . If W is a subspace
of V than contains S, then W contains Span(S).
Proposition 2. Let S be a subset of V . If x is in Span(S), then Span(S ∪ {x}) = Span(S).
Proposition 3. Consider a set of vectors {x1 , . . . , xn } in V . The following are equivalent:
(1) The set of vectors is linearly dependent.
(2) There is an i such that xi is a linear combination of {x1 , . . . , xi−1 , xi+1 , . . . , xn }.
(3) There is an i such that xi is a linear combination of {x1 , . . . , xi−1 }.
Proposition 4. Let S1 ⊆ S2 be subsets of V .
(1) If S1 is linearly dependent, then S2 is linearly dependent.
(2) If S2 is linearly independent, then S1 is linearly independent.
Theorem 5. Let β = {x1 , . . . , xn } be a finite set of V . Then β is a basis of V if and only if
every element x of V can be written uniquely as a linear combination
x = a1 x 1 + . . . + an x n .
Theorem 7 (Replacement Theorem). Assume that V has a finite generating set S with n
elements. Let L be a linearly independent set of V /. Then L has finitely many elements,
say m, and m ≤ n. Moreover, there exists a subset H of S containing n − m vectors, such
that L ∪ H generates V .
Corollary 7.2. Let V be an n-dimensional vector space.
(1) If S is a spanning set of V with m elements, then m ≥ n and there exists an n element
subset of S which is a basis of V .
(2) If L is a linearly independent subset of V with k elements, then k ≤ n and there
exists an n − k element subset H of V such that L ∪ H is a basis of V .
Corollary 7.3. Let V be an n-dimensional vector space and S = {x1 , . . . , xn } a subset.
(1) If S spans V , then S is a basis.
(2) If S is linearly independent, then S is a basis.
Corollary 7.4. Let V be a finite dimensional vector space and W ≤ V a subspace of V .
Then W is finite dimensional, dim W ≤ dim V , and dim W = dim V if and only if W = V .
Theorem 10 (Rank-Nullity Theorem). Let T : V → W be a linear map and assume that
V is finite dimensional. Then
null T + rank T = dim V.
1
Theorem 11. Let V, W be vector spaces, and assume V is finite dimensional. Let β =
{x1 , . . . , xn } be a basis of V . For any vectors w1 , . . . , wn of W , there exists a unique linear
map T : V → W such that
T (xi ) = wi
for i = 1, . . . , n.
Theorem 12. Let T : V → W and U : W → Z be linear transformations. Let α, β and γ
be bases of V, W and Z, respectively. Then
[U ◦ T ]γα = [U ]γβ [T ]βα .
Corollary 12.1. Let T : V → W be a linear transformation. Let α, β be bases of V and
W , respectively. Then, for x ∈ V , we have
[T (x)]β = [T ]βα [x]α .
In particular, if T = 1V , then we have
[x]β = [1V ]βα [x]α .
Proposition 13. Let T : V → W be a linear transformation and assume that dim V =
dim W . Then the following are equivalent:
(1) T is an isomorphism;
(2) there exists a linear map T −1 : W → V such that T −1 ◦ T = 1V and T ◦ T −1 = 1W ;
(3) T is onto;
(4) T is 1-1;
(5) N (T ) is zero.
(6) rank T = dim V .
Theorem 14. Let V, W be finite dimensional vector spaces. Then V is isomorphic to W if
and only if dim V = dim W .
Theorem 15. Let V, W be finite dimensional vector spaces with bases β and γ, respectively.
Assume dim V = n and dim W = m.
(1) The map φβ : V → F n is an isomorphism.
(2) Let T : V → W be a linear transformation and let A = [T ]γβ . The following diagram
is commutative.
T
/W
V
φβ ∼
=
Fn
∼
= φγ
LA
/
Fm
Theorem 20. Let T : V → V , where V is a finite dimensional vector space. If v1 , . . . , vk
are eigenvectors with eigenvalues λ1 , . . . , λk such that λi 6= λj for all i 6= j, then v1 , . . . , vk
are linearly independent in V .
Corollary 20.1. Let T : V → V , where V is a finite dimensional vector space. Let λ1 , . . . , λk
be the eigenvalues of T with eigenspaces E1 , . . . , Ek . If Si is a linearly independent subset
of Ei for i = 1, . . . , k, then S = S1 ∪ . . . ∪ Sk is a linearly independent set of V .
Theorem 24 (Diagonalization Theorem). Let T : V → V , where V is a finite dimensional
vector space. Then T is diagonalizable if and only if fT (t) splits and dim Eλ = m(λ) for
every eigenvalue λ.
Theorem 25. Let V be an inner product space and let S = (x1 , . . . , xn ) be an orthonormal
set of vectors in V . If y ∈ Span(S), then
y = hy, x1 ix1 + . . . + hy, xn i.
In particular, if β = (x1 , . . . , xn ) is an orthonormal basis of V , then for any y in V , we have


hy, x1 i
..
.
[y]β = 
.
hy, xn i
Theorem 26 (Gram-Schmidt). Let V be an inner product space and let S = (w1 , . . . , wn )
be a linearly independent set. Define S 0 = (v1 , . . . , vn ), where v1 = w1 and for 2 ≤ k ≤ n,
set
k−1
X
hwk , vj i
vk = wk −
· vj .
||vj ||2
j=1
Then S 0 is an orthogonal set with Span(S) = Span(S 0 ).
Theorem 27. Let W be a finite dimensional subspace of an inner product space V . Then
for every x in V , there is a unique w in W and a unique y in W ⊥ such that
x = w + y.
Corollary 27.1. Let W be a finite dimensional subspace of an inner product space V . There
exists a linear transformation
projW : V → V
such that R(projW ) = W , N (projW ) = W ⊥ and projW (x) = x for all x ∈ W .
Corollary 27.2. Let V be a finite dimensional inner product space and W a subspace.
Then
dim W + dim W ⊥ = dim V.
Theorem 28. Let V be an inner product space over R. Define a function
ψ :V →Vd
by ψ(y)(x) = hx, yi (so ψ(y) : V → R1 ). Then:
(1) The function φ is a linear transformation;
(2) φ is 1-1;
(3) if V is finite dimensional, then φ is an isomorphism.
Corollary 28.1. Let V be a finite dimensional inner product space over R, let φ : V → R1
be a linear transformation. Then there exists an element y of V such that
φ(x) = hx, yi
for all x in V .
From this point on we assume that V is a finite dimensional inner product space over R.
Theorem 29. Let T : V → V be a linear operator. There exists a unique liner operator
T ∗ : V → V such that
hT (x), yi = x, T ∗ (y)i
for all x, y in V .
Proposition 30. Let T : V → V be a linear operator and let β be an orthonormal basis of
V . Then [T ∗ ]β = ([T ]β )tr .
Theorem 31. Let T : V → V be a linear operator. Then T is symmetric if and only if
there is an orthonormal basis of eigenvectors of T .
Corollary 31.1. Let A be an n × n matrix of real numbers. Then A is symmetric if and
only if there is an orthogonal matrix Q such that QT AQ = D, where D is a diagonal matrix.
Theorem 32. Let T : V → V be a linear operator. The following are equivalent.
(1) T T ∗ = 1V ;
(2) hT (x), T (y)i = hx, yi for all x, y in V ;
(3) for any orthonormal basis β of V , T (β) is an orthonormal basis of V ;
(4) there exists an orthonormal basis of V such that T (β) is an orthonormal basis of V ;
(5) ||T (x)|| = ||x|| for all x in V .
Corollary 32.1. T is orthogonal if and only if T is an isomorphism and T −1 = T ∗ .
Corollary 32.2. Let A be an n × n matrix of real numbers. The following are equivalent.
(1) AAT = In ;
(2) the columns of A are orthonormal;
(3) ||Ax|| = ||x|| for all x in Rn .
Lemma 33. Let A be an m × n matrix of real numbers. Then:
(1) The n × n matrix AT A has n eigenvalues, counted with multiplicity, and they are all
non-negative.
(2) Let x1 , . . . , xn be an orthonormal basis of eigenvectors for AT A with eigenvalues λi .
Then ||Axi || = λ2i .
(3) hAxi , Axj i = 0 for i 6= j.
Theorem 34. Let A be an m × n matrix of real numbers, let σ1 , . . . , σr be the nonzero
singular values of A. Let Σ = [sij ] be the m × n matrix with sii = σi for i = 1, . . . , r and
sij = 0 otherwise. Then there exists an m × m orthogonal matrix U and an n × n orthogonal
matrix V such that
A = U ΣV T .