3X4 matrix is row equivalent to 1 0 3 0 0 1 2 0 0 0 0 1 a Is

3X4 matrix is row equivalent to

[1 0 -3 0]
[0 1   2 0]
[0 0   0 1]

a. Is the linear transformation x--> Ax onto (why?)

b. Is the linear transformation x--> Ax one-to-one (why?)

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Solution

The corresponding linear transformation x-> Ax of this matrix will be from R⁴ to R³. So the linear transformation can not be one one because the dimenstion of domain (4) is greater than dimension of codamain (3).

The second, third and fourth columns { {0,1,0}, {-3,2,0}, {0,0,1}} are linearly independent so they can generate R³.

Since the column space generates R³, so the linear transformation will be onto.

So the corresponding linear transformation x-> Ax is onto but not one-to-one.