⎝⎛x2x22x4+x5x4x5⎠⎞=x2⎝⎛1100⎠⎞+x4⎝⎛00210⎠⎞+x5⎝⎛00101⎠⎞
Matrix S=⎝⎛110000021000101⎠⎞
rref ofS=⎝⎛100000100000100⎠⎞
The three vectors are linearly independent and they are the basis of S
Dimension of S=3
PART 2
⎝⎛x+yx−yx+2z⎠⎞=x⎝⎛111⎠⎞+y⎝⎛1−10⎠⎞+z⎝⎛002⎠⎞
Matrix T=⎝⎛1111−10002⎠⎞
rref of T=⎝⎛100010001⎠⎞
The vectors are linearly independent.
The basis of the range T is
[⎝⎛111⎠⎞,⎝⎛1−10⎠⎞,⎝⎛002⎠⎞]
3, Part1
Every linear transformation is associated with a matrix
(u−v+2w5v−w)=u(10)+v(−15)+w(2−1)
T1=(10−152−1)
⟹ T1 is linear
3) Part2
∫abk2p(x)dx=k∫ab2p(x)dx
∫ab2p(x1+x2)dx
=∫ab2p(x1)dx+∫ab2p(x2)dx
Therefore T2 is linear
3). Part 3
T3P(u1+u2)=(u1+u2)P(u1+u2)+(u1+u2)
T3P(u1)=u1P(u1)+u1
T3P(u2)=u2P(u2)+u2
∴T3P(u1+u2)/=T3P(u1)
+T3P(u2)
Therefore T3 is not linear
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