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MT210 TEST 2
İLKER S. YÜCE
APRIL 19, 2011
SURNAME, NAME:
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 − x 2 + 2 x 3 −x 1 − 2 x 2 + x 3 2 x 1 + x 2 + x 3
a.) Find the standard matrix of T. b.) Is T one-to-one? Explain. c.) Is T onto? Explain.
d.) If there is any, find a vector v ⃗ 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 solutions other than the trivial solution. c) No, T is not onto. Because, the number of pivot positions is less than the number of rows. d) Yes, there is such vector, e.g., v ⃗ =
(^53) −^13 0
, because we have
[ A⃗ b ] =
0 1 − 1 −^13
QUESTION 2. MATRIX OPERATIONS
Define the linear transformations T : R^4 Ï R^2 and S : R^2 Ï R^3 so that
T
x 1 x 2 x 3 x 4
[ (^) x 1 −^ x 2 +^ x 3 +^ x 4 x 1 + x 2 + x 3 − x 4
]
and S
([ (^) x 1 x 2
])
−x 1 + x 2 x 1 − 2 x 2 2 x 1 − x 2
a.) Find the standard matrix of S ◦ T. b.) Is it possible to mention T ◦ S? Explain.
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 =
[ 1 − 1 1 1
]
, B =
Ñ BA =
(b.) No, it is not possible to mention T ◦ S because codom ( S ) ̸ = dom ( T ). (c.) There is no such v⃗ because
[ BA⃗ b ] =
QUESTION 4. CHARACTERIZATIONS OF INVERTIBLE MATRICES
Use the invertible matrix theorem to determine the value(s) of λ for which the matrix
A =
1 λ 0 3 2 0 1 2 1
is NOT invertible.
ANSWER
We reduce the given matrix:
A ∼
1 λ 0 3 2 0 1 2 1
∼ B =
1 λ 0 0 2 − 3 λ 0 0 − 4 − 3
By IMT, A is not invertible if and only if the columns of A are linearly dependent. Assume that λ ̸ = 23. Then we see that
B ∼
1 λ 0 0 2 − 3 λ 0 0 − 4 − 3
∼ C =
1 λ 0 0 2 − 3 λ 0 0 0 − 3
Note that C has 3 pivot position as long as λ ̸ = 23. Therefore we conclude that A is not invertible if and only if 2 − 3 λ = 0 or λ = 23.
QUESTION 5. SUBSPACES OF R n
Define the linear transformation T : R^4 Ï R^3 by T
x 1 x 2 x 3 x 4
x 1 + x 2 x 2 − x 3 x 1 + x 4
a.) Find the column space of T. Find the dimension of the column space. b.) Find the null space of T. Find the dimension of the null space. c.) Find a basis for the column space of T. d.) Find a basis for the null space of T .(Remark. The column space of T is Col A and the null space of T is Nul A where A is the standard matrix of T .)
ANSWER
(a) The standard matrix A of T is
A =
(^) Ñ Col A =
c^1
(^) + c 2
(^) + c 3
(^) + c 4
(^) : c 1 , c 2 , c 3 , c 4 ∈ R
(b)
A =
(^) Ñ Nul A =
^ x^4
:^ x^4 ∈^ R
(c) A =
Ñ BC =
(d)
Nul A =
^ x^4
:^ x^4 ∈^ R
^ Ñ BN^ =
^.
