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Let A, B, C and D be the following matrices:\begin{align*} &\mathbf{A} \text{ sends every vector to the zero vector}, \\ &\mathbf{B} \text{ reflects vectors across the xz-plane}, \\ &\mathbf{C} \text{ projects vectors onto the vector \mathbf{i} + \mathbf{j}}, \\ &\mathbf{D} \text{ sends every vector to twice itself}. \end{align*}

For each matrix in the list above, figure out whether the matrix is invertible, then enter in "invertible" or "not invertible".

For each matrix in the list above, enter in "point" if the output grid covers a point, "line" if the output grid covers a line, "plane" if the output grid covers a plane, and "3-space" if it covers all of three-dimensional space.

Aug 6, 2019

#1
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the way they want you to look at this problem is to determine whether the operations listed can be reversed to

obtain the original value.

For example to reverse D we find a matrix that sends every vector to 1/2 of itself.

See if you can apply this thinking to the others.

Aug 6, 2019
#2
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oh I get the first part now. what about the second part?

Aug 6, 2019
#3
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well basically you are looking at the set of output from each matrix when fed all possible vectors.

The matrix being invertible should immediately suggest one of the 3 options.

Think about the output of matrix A.  It tells you it maps every vector to the zero vector.

Is the zero vector a point, line, plane, or all of 3D space?

Simlarly with C.  Every output it produces lies along the vector i+j.  That suggests that the output grid covers a .. ?

Rom  Aug 7, 2019