Linear Algebra Questions and Answers for Freshers focuses on “Rank of Matrix in Row Echelon Form”.
1. Write Matrix corresponding to the following linear transformations.
y1=2x1-x2-x3
y2=3x3
y3=x1+x2
a) (begin{bmatrix}2&-1&-1\0&0&3\1&1&0end{bmatrix})
b) (begin{bmatrix}2&-1&-1\0&0&3\1&0&0end{bmatrix})
c) (begin{bmatrix}2&-1&-1\0&0&3\1&1&1end{bmatrix})
d) (begin{bmatrix}2&-1&-1\0&1&3\1&1&0end{bmatrix})
Answer: a
Explanation: In the given question,
We know that Linear Transformation is given by,
(begin{bmatrix}y1\y2\y3end{bmatrix}begin{bmatrix}2&-1&-1\0&0&3\1&1&0end{bmatrix}begin{bmatrix}x1\x2\x3end{bmatrix})
Thus, the matrix for linear transformation is
(begin{bmatrix}2&-1&-1\0&0&3\1&1&0end{bmatrix}).
2. Which of the following Linear Transformations is not correct for the given matrix?
(begin{bmatrix}1&2&-3\-1&0&1\2&1&0end{bmatrix})
a) x1=1y1-3y2-3y3
b) x1=1y1-2y2-3y3
c) x2=-1y1+1y3
d) x3=2y1+y2
Answer: a
Explanation: In the given question,
X=(begin{bmatrix}1&2&-3\-1&0&1\2&1&0end{bmatrix})Y
Thus,
x1=1y1-2y2-3y3
x2=-1y1+1y3
x3=2y1+y2.
3. For the linear transformation, X=(begin{bmatrix}2&1&1\1&1&2\1&0&-2end{bmatrix})Y, find the Y co-ordinates for (1, 2, -1) in X.
a) (0, -2, 0)
b) (-1, 3, 1)
c) (-1, -2, 0)
d) (-1, 3, 0)
Answer: d
Explanation: In the given question,
X=(1, 2, -1)
(begin{bmatrix}1\2\-1end{bmatrix}begin{bmatrix}2&1&1\1&1&2\1&0&-2end{bmatrix}begin{bmatrix}y1\y2\y3end{bmatrix})
y1– 2y3=-1
y2+4y3=3
y3=0
Thus Y (-1, 3, 0).