Linear Algebra, 2020a

266 Chapter Three. Maps Between Spaces
(a)
( ) ( ) ( ) ( )
5 0 2 1 −1 4
1 −1
(b)
(c)
(d)
0 4 3 1
2 −8 1 1
1.13 For each space find the matrix changing a vector representation with respect
to B to one with respect to D.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 0
(a) V = R 3 , B = E 3 , D = 〈 ⎝2⎠ , ⎝1⎠ , ⎝ 1 ⎠〉
3 1 −1
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 0
(b) V = R 3 , B = 〈 ⎝2⎠ , ⎝1⎠ , ⎝ 1 ⎠〉, D = E 3
3 1 −1
(c) V = P 2 , B = 〈x 2 ,x 2 + x, x 2 + x + 1〉, D = 〈2, −x, x 2 〉
1.14 Find bases such that this matrix represents the identity map with respect to
those bases.
⎛
3 1
⎞
4
⎝2 −1 1⎠
0 0 4
1.15 Consider the vector space of real-valued functions with basis 〈sin(x), cos(x)〉.
Show that 〈2 sin(x)+cos(x),3cos(x)〉 is also a basis for this space. Find the change
of basis matrix in each direction.
1.16 Where does this matrix (cos(2θ) sin(2θ)
)
sin(2θ) −cos(2θ)
send the standard basis for R 2 ? Any other bases? Hint. Consider the inverse.
̌ 1.17 What is the change of basis matrix with respect to B, B?
1.18 Prove that a matrix changes bases if and only if it is invertible.
1.19 Finish the proof of Lemma 1.5.
̌ 1.20 Let H be an n×n nonsingular matrix. What basis of R n does H change to the
standard basis?
̌ 1.21 (a) In P 3 with basis B = 〈1+x, 1−x, x 2 +x 3 ,x 2 −x 3 〉 we have this representation.
⎛ ⎞
0
Rep B (1 − x + 3x 2 − x 3 )= ⎜1
⎟
⎝1⎠
2
B
Find a basis D giving this different representation for the same polynomial.
⎛ ⎞
1
Rep D (1 − x + 3x 2 − x 3 )= ⎜0
⎟
⎝2⎠
0
D
(b) State and prove that we can change any nonzero vector representation to any
other.
Hint. The proof of Lemma 1.5 is constructive — it not only says the bases change,
it shows how they change.
1.22 Let V, W be vector spaces, and let B, ˆB be bases for V and D, ˆD be bases for
W. Where h: V → W is linear, find a formula relating Rep B,D (h) to RepˆB, ˆD (h).