Question 8 Let T: R4 R4 be the matrix transformation Incomplete 1 -1 2 1 answer

Question

Question 8 Let T: R4 R4 be the matrix transformation Incomplete 1 -1 2 1 answer 2 2 3 2 Marked out of T 1 0 10.00 1 -4 5 h Flag question Answer the following questions TRUE or FALSE. Justify your answers. Choose the most appropriate answer!!! T is 1-1 when h 5 The kernel of T consists only of the zero vector when h T is 1-1 when h range of Tequals R4 when h 2. T is onto when h 3 The kernel of T contains non-zero vectors when h 2. Please answer all parts of the question Choose Choose Choose Choose False Choose

Explanation / Answer

Let the given matrix be denoted by A. We will reduce A to its REREF as under:

Add -2 times the 1st row to the 2nd row

Add 2 times the 1st row to the 3rd row

Add -1 times the 1st row to the 4th row

Interchange the 2nd row and the 3rd row

Multiply the 2nd row by -1/3

Add 3 times the 2nd row to the 4th row

Multiply the 3rd row by -1

Add 1 times the 3rd row to the 4th row

Add 4/3 times 3rd row to 2nd row

Add 1 time 2nd   row to the 1st   row

Add -2 times 3rd row to the 1st row

Then the RREF of A is

1

0

0

2/3

0

1

0

-1/3

0

0

1

0

0

0

0

h-2

Multiply the 4th row by 1/3

Add -2/3 times the 4th row to the 1st row

Add 1/3 times the 4th row to the 2nd row.

Then the RREF of A changes further to I4.

This means that the columns of A are linearly independent. Hence T is 1-1.The statement is True

2. When h = 3, then also, the RREF of is I4 . Then Ker(T) = 0. The statement is True.

3. When h =2, the 4th row of the RREF of A is (0,0,0,0). Then the 4th column of A is a linear combination of its 1st and 2nd columns. In this case T cannot be 1-1. The statement is False.

4. When h 2, the RREF of A is I4. Then the columns of A span R4. Now, since the Range of T is same as Col(A) = R4, the statement is True.

4. When h = 3, then also, the RREF of is I4 . Then the columns of A span R4. Hence T is onto.

5. When h =2, the 4th row of the RREF of A is (0,0,0,0).Then Ker(T) = Ker(A) contains vectors whose 4th row is arbitrary. These vectors can be non-zero. The statement is True.

1

0

0

2/3

0

1

0

-1/3

0

0

1

0

0

0

0

h-2

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