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Practice Problems
- Find a basis for the span(S) as a subspace of R^3 where
(a) S =
(b) S =
- Find a basis for the subspace
S = {x ∈ R^4 |Ax = 0 } of R^4 where A =
- Determine whether the following set is a basis for M 2 × 2 :
B =
- Show that B = {x + 1, x − 1 , x^2 } is a basis for P 2. Write the vector p(x) = 8x^2 + x + 5 as a unique linear combination of the vectors from the set B.
- Show that if S = {v 1 , v 2 ,... , vn} is a basis for the vector space V and c is a nonzero scalar, then S′^ = {cv 1 , cv 2 ,... , cvn} is also a basis for V.
- Show that if S = {v 1 , v 2 ,... , vn} is a basis for Rn^ and A is an invertible matrix, then S′^ = {Av 1 , Av 2 ,... , Avn} is also a basis.
- Find dim(W ), where
W =
2 s + t + 3r 3 s − t + 2r s + t + 2r
(^) s, t, r ∈ R
- Let S = {v 1 , v 2 , v 3 }, where
v 1 =
v^2 =
v^3 =
(a) Explain why S is not a basis for R^4. (b) Show that v 3 is a linear combination of v 1 and v 2. (c) Find the dim(span(S))). (d) Find a basis B for R^4 that contains v 1 and v 2.
(e) Show that
T =
is a basis for R^4.
- Find a basis for R^4 containing the vectors
- Find a basis for the column space of A =
, then find a basis for the null space of the same matrix. What is the rank of A?
- Determine whether the function is a linear transformation between the given vector spaces:
(a) T : R^2 → R, T
x y
= x^2 + y^2
(b) T : R^3 → R^3 , T
x y z
x + y − z 2 xy x + y + 1
(c) T : P 3 → R^2 , T (ax^3 + bx^2 + cx + d) =
−a − b + 1 c + d
(d) T : M 2 × 2 → M 2 × 2 defined by T (A) =
A.
- Let B 1 =
v^1 =
(^) , v 2 =
(^) , v 3 =
and^ B^2 =
u^1 =
(^) , u 2 =
(^) , u 3 =
be two ordered bases for R^3.
(a) Find the coordinates of w 1 =
(^) with respect to each basis.
(b) Find the change of basis matrix [A]B 1 →B 2 from B 1 to B 2. (c) Use [A]B 1 →B 2 to check your answer from part (a).
(d) Find the vector w 2 with [w 2 ]B 1 =
(e) Compute [A]B 2 →B 1 from B 2 to B 1 and use it to compute [w 2 ]B 2.
- Suppose that B 1 = {u 1 , u 2 , u 3 } and B 2 = {v 1 , v 2 , v 3 } are ordered bases for a vector space R^3 such that u 1 = −v 1 + 2v 2 , u 2 = −v 1 + 2v 2 − v 3 , and u 3 = −v 2 + v 3
- Use the vectors
to ”build” a basis for M 2 × 2.
- Find bases for the null space N ull(A) and the column space Col(A) of the matrix
A =
. Indicate the dimensions of these vector spaces.
- Prove that the function T : M 2 × 2 → P 2 defined by T
a b c d
= bx^2 + cx − a is a linear transformation.
- Let T : R^2 → R^2 be given by:
T
x y
x + y x − y
(a) Show that T is a linear map. (b) Find a basis for the nullspace of T. (c) Find the image of each of the standard basis vectors under T.
- Let T : R^3 → R^3 be a linear operator given by T
, T
, T
and let B =
(a) Show that B is a basis for R^3. (b) Find [T ]B→B.
(c) Find T
(^) using the matrix found in the previous part.
- Suppose T 1 : V → R and T 2 : V → R are linear transformations. Define T : V → R^2 by
T (v) =
T
1 (v) T 2 (v)
for v ∈ V. Show that T is a linear transformation.
- Suppose that T 1 : V → V and T 2 : V → V are linear operators on V and {v 1 , v 2 ,... vn} is a basis for V. Show that if T 1 (vi) = T 2 (vi) for i = 1, 2 ,... n, then T 1 (v) = T 2 (v) for all v ∈ V.
- Let S = {v 1 , v 2 , v 3 } be a linearly independent subset of R^3. Find a linear operator T : R^3 → R^3 such that {T (v 1 ), T (v 2 ), T (v 3 )} is linearly dependent.
- Recall that the trace (denoted tr( )) of an n × n matrix A is the sum of the entries on its diagonal. Define T : Mn×n → R by T (A) = tr(A). Is T a linear transformation? If so, is it injective, surjective, both or neither? Justify.
- Define T : R^4 → R^3 by
T
x y z w
x + y − z + w 2 x + y + 4z + w 3 x + y + 9
Find a basis for null(T).
- Let T : P 2 → P 2 be defined by
T (ax^2 + bx + c) = (a + b)x^2 + cx + (a + b). Find a basis for range(T).
- Let T : R^2 → R^3 be the linear transformation defined by
T
x 1 x 2
x 2 x 1 + x 2 x 1 − x 2
Let B =
and B′^ =
be ordered bases for^ R
(^2) and R (^3) respec- tively. (a) Find [T ]B→B′ (b) Find T
using part (a) and directly.
- Let D : P 3 → P 3 be the linear operator defined by
D(p(x)) = p′(x). Find the matrix of D relative to the standard basis B = { 1 , x, x^2 , x^3 }, and use it to find the derivative of p(x) = 1 − x + 2x^3.
