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MT210 TEST 2 SAMPLE 2
ILKER S. YUCE
MARCH 29, 2011
QUESTION 1. THE MATRIX OF A LINEAR TRANSFORMATION
Define the linear transformation T : R^3 Ï R^3 so that
x 1 x 2 x 3
7 Ï
x 1 + 2 x 3 2 x 1 + x 2 + 3 x 3 x 1 − x 2 + 3 x 3
a.) Find the standard matrix of T. b.) Is T one-to-one? c.) Is T onto? d.) If there is any, find a vector v⃗
such that Tv ( ⃗ ) = ⃗b where b⃗ =
ANSWER
(a.) The standard matrix is A where
A =
(b) No, T is not one-to-one. Because A⃗x = ⃗ 0 has infinitely many solutions. (c) No, T is not onto. Because, the number of pivot positions is not equal to the number of rows. (d) No, there is no such vector because we have
[ A⃗ b ] =
QUESTION 2. MATRIX OPERATIONS
Define the linear transformations T : R^3 Ï R^3 and S : R^3 Ï R^3 so that
T
x 1 x 2 x 3
x 1 − 2 x 2 + x 3 x 2 − x 3 2 x 3
(^) and S
x 1 x 2 x 3
−x 3 −x 2 + 2 x 3 −x 1 + 2 x 2 − x 3
a.) Find the standard matrix of S ◦ T. b.) Find the standard matrix of T ◦ S. c.) Find, if there is any, a
vector v⃗ such that ( S ◦ T )( v⃗ ) = ⃗b where b⃗ =
ANSWER
(a.) The standard matrix BA of S ◦ T is the product of the standard matrix B of S and the standard matrix A of T where
A =
, B =
Ñ BA =
(b.) The standard matrix AB of T ◦ S is the product of the standard matrix B of S and the standard matrix A of T where
A =
, B =
Ñ AB =
(c.) [ BA⃗ b ] =
(^) Ñv⃗ =
QUESTION 4. CHARACTERIZATIONS OF INVERTIBLE MATRICES
Use the invertible matrix theorem to determine the value(s) of λ for which the matrix
λ − 1 0 − 1 λ − 1 0 − 1 λ
is invertible.
ANSWER
Let us assume that λ ̸ = 0. Then we get
A =
λ − 1 0 − 1 λ − 1 0 − 1 λ
− 1 λ − 1 0 − 1 λ 0 0 λ ( λ^2 − 1) − λ
We solve λ ( λ^2 − 1) − λ = 0 and we derive that λ ̸ = 0 and λ ̸ =
2 and λ ̸ = −
[ A|I ] ∼
^10 01
−^ − 2 1+ λ + λλ^23 − 2 λ^ λ + λ^3 − 2 λ^1 + λ^3 − 2 λ^ λ + λ^3 − 2^ λλ^2 + λ^3 − 2 λ^ λ + λ^3 − 2 λ^1 + λ^3 − 2 λ^ λ + λ^3 −^ − 2 1+ λ + λλ^23
Ñ A− 1 =
−^ − 2 1+ λ + λλ^23 − 2 λ^ λ + λ^3 − 2 λ^1 + λ^3 − 2 λ^ λ + λ^3 − 2^ λλ^2 + λ^3 − 2 λ^ λ + λ^3 − 2 λ^1 + λ^3 − 2 λ^ λ + λ^3 −^ − 2 1+ λ + λλ^23
QUESTION 5. SUBSPACES OF R n
Let T : R^3 Ï R^3 be a linear transformation and B = {v⃗ (^) 1 v,⃗ (^) 2 v,⃗ (^) 3 } a basis for R^3. Suppose Tv ( ⃗ (^) 1 ) = ( − 2 , 1 , 1), Tv ( ⃗ (^) 2 ) = (0 , 1 , − 1) and Tv ( ⃗ 3 ) = ( − 2 , 2 , 0). a.) Determine whether w⃗ = ( − 6 , 5 , 0) is in the range of T. b.) Find a basis for the range of T. c.) Find a basis for the null space of T .(Remark. Range of T is the same space as the column space of A where A is the standard matrix of T .)
ANSWER
(a) We need to solve x 1 Tv ( ⃗ (^) 1 ) + x 2 Tv ( ⃗ (^) 2 ) + x 3 Tv ( ⃗ (^) 3 ) = w⃗ :
Therefore, w⃗ is not in the range. (b) Range T=Col A where A = [ Tv ( ⃗ (^) 1 ) Tv ( ⃗ 2 ) Tv ( ⃗ (^) 3 )]. Thus, we get
BC =
(c) Nul A =
x^3
(^) : x 3 ∈ R
. Thus, we get
BN =
