"Determinant"-Graphical linear algebra 04

This paper turns from the public number---meets the mathematical---graphic mathematical---linear algebra part

Thank you for meeting the Math Working Group to explain the obscure and acting knowledgeable mathematical knowledge of university textbooks in an easy-to-understand and lively and interesting way.

This time we mainly do a review, and then further the determinant of the geometric meaning of the animation to show the explanation. We say that matrix A can be considered a linear transformation, so

The above equation means that a vector x is coincident with the vector v at the position after the linear transformation a . Now look at an example of how the whole space changes under the action of matrix A:

• The original vector (1, 0.5) after the transformation is (2, 1.5);

• The horizontal direction becomes twice times the original;

• The longitudinal becomes 3 times times the original;

• The original line is still straight after the transformation, parallel remains parallel;

• The origin does not change (if there is no origin, it is the affine space).

And notice that the Red block area is enlarged 6 times times, such an area (or volume) magnification is the geometric meaning of the determinant (determinant), recorded as: Det (A) or | a|

Look at another animation of the linear transformation of the function matrix:

Observed to see:

• Space has been tilted, but not distorted;

• The straight line is still straight, parallel remains parallel;

• The first column (1.5,-1) of a has a foothold of (1, 0)-like, the second column (-0.5, 2) at the point of (0, 1);

• The unit red Small square expands to 2.5 times times, namely det (A) = 2.5

Looking at this example of linear transformation, note that the two-column vectors in matrix A are proportional-linearly related:

Observations were:

• The space is compressed into a line;

• The vector (1, 0.5) does not change at all during the entire transformation process (this is related to the eigenvalues and eigenvectors, and we post-instrument again);

• The area magnification is 0, i.e. det (A) = 0;

This is similar to the one in the previous section where the matrix diagonal contains 0 elements, in which case it means there is no inverse matrix, but it is something to be introduced later.

Determinant of the geometric meaning of the area (volume) of the increase in the magnification, such as after the mirror rollover is negative, the previous section we see the three-dimensional matrix of the situation, now look at the two-dimensional image after the reversal of the determinant of the change, note that the Det (A) value from positive to negative changes in the process:

"Determinant"-Graphical linear algebra 04