Linear Combination Mapping: Is the Invertible Matrix Theorem True or False?

[henry3369]

True or False:
If the linear combination x -> Ax maps Rⁿ into Rⁿ, then the row reduced echelon form of A is I.

I don't understand why this is False. My book says it is false because it is only true if it maps Rⁿ ONTO Rⁿ instead of Rⁿ INTO Rⁿ. What difference does the word into make?

 

[Mark44]

True or False:
If the linear combination x -> Ax maps Rⁿ into Rⁿ, then the row reduced echelon form of A is I.

I don't understand why this is False. My book says it is false because it is only true if it maps Rⁿ ONTO Rⁿ instead of Rⁿ INTO Rⁿ. What difference does the word into make?

Consider the transformation whose matrix looks like this:
$$A =
\begin{bmatrix}1 & 0 & 0 & \dots & 0 \\
0 & 0 & 0 & \dots & 0 \\
\vdots \\
0 & 0 & 0 & \dots & 0
\end{bmatrix}$$
This transformation maps a vector x to its projection on the $x_1$ axis, a map from Rⁿ into Rⁿ (but not onto Rⁿ). Does it seem likely to you that this matrix will row reduce to the identity matrix?

 

[henry3369]

Consider the transformation whose matrix looks like this:
$$A =
\begin{bmatrix}1 & 0 & 0 & \dots & 0 \\
0 & 0 & 0 & \dots & 0 \\
\vdots \\
0 & 0 & 0 & \dots & 0
\end{bmatrix}$$
This transformation maps a vector x to its projection on the $x_1$ axis, a map from Rⁿ into Rⁿ (but not onto Rⁿ). Does it seem likely to you that this matrix will row reduce to the identity matrix?

So if it maps onto Rⁿ does that mean that it maps x into every position in Rⁿ?

 

[henry3369]

Also can you clarify why these two statements are equivalent in the Invertible Matrix Theorem::
1. The equation Ax = b has at least one solution for each b in Rⁿ
2. The linear transformation x -> Ax is one-to-one.

What is throwing me off is the word least in statement 1. Shouldn't it have exactly one solution for every b in order for the transformation to be one-to-one?

 

[Mark44]

Also can you clarify why these two statements are equivalent in the Invertible Matrix Theorem::
1. The equation Ax = b has at least one solution for each b in Rⁿ
2. The linear transformation x -> Ax is one-to-one.

What is throwing me off is the word least in statement 1. Shouldn't it have exactly one solution for every b in order for the transformation to be one-to-one?

Yes, I'm bothered by it as well. It's technically correct, but misleading, as it seems to imply that for some b there might two input x values. That can't happen if the transformation is one-to-one, though.

To me, it's sort of like saying 3 + 4 is at least 7.

 

[WWGD]

Yes, this is rank-nullity, together with the fact that a linear map is injective iff it has a trivial kernel.