Practice: Linear transformations as moves
Self-check
Section titled “Self-check”Six short questions. Answer each one in your head (or on paper) before opening the collapsible. Trying to retrieve the answer is where the learning sticks; rereading feels productive but does much less.
1. What two requirements make a transformation linear?
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The origin stays fixed (the zero vector maps to the zero vector), and straight lines stay straight (grid lines stay parallel and evenly spaced; no curving, no uneven bunching). Stretch, rotate, shear, and flip are allowed; curving and shifting the whole plane are not.
2. Why is knowing where i-hat and j-hat land enough to know what the transformation does to every vector?
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Because every vector is a linear combination v = x · i-hat + y · j-hat, and a linear transformation preserves that combination: L(v) = x · L(i-hat) + y · L(j-hat). The same coordinates x and y now scale the transformed basis vectors, so the two landing spots reconstruct every destination.
3. What do the two columns of a 2x2 matrix represent?
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The first column is where i-hat lands, and the second column is where j-hat lands. A matrix is just those two transformed basis vectors written side by side as columns.
4. What does the matrix-vector product M · v compute, in terms of the columns?
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For v = [x, y], it is x · (first column) + y · (second column), which is exactly x · L(i-hat) + y · L(j-hat). It is the linear-combination idea reassembling your vector out of the transformed basis vectors, not an arbitrary memorized rule.
5. To sketch what a matrix does to the unit square, what do you actually draw?
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Plot the two columns as arrows from the origin (where i-hat and j-hat land), then draw the parallelogram they span. That parallelogram is the image of the unit square and shows the whole effect: stretch, rotation, shear, flip, or collapse.
6. What is the difference between a linear transformation and an affine one?
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A linear transformation must keep the origin fixed. An affine transformation can also shift the whole plane by a constant amount (a translation). A bare 2x2 matrix can only do linear transformations; the fixed shift needs an extra added vector.
Try it yourself, part 1: build the matrix, then apply it
Section titled “Try it yourself, part 1: build the matrix, then apply it”For each described transformation, first write its 2x2 matrix (column 1 is where i-hat lands, column 2 is where j-hat lands), then apply the matrix to the test vector v = [2, 3] using x · col1 + y · col2. About 8 minutes, pen and paper.
- a) A vertical stretch by 3 (
i-hatunchanged,j-hattripled). - b) A 90-degree clockwise rotation.
- c) A reflection across the x-axis (flip top and bottom).
- d) A shear that leaves
i-hatat[1, 0]and sendsj-hatto[2, 1].
Check your work
- a)
i-hatto[1, 0],j-hatto[0, 3]. Matrix[[1, 0], [0, 3]]. Apply:2 · [1, 0] + 3 · [0, 3] = [2, 0] + [0, 9] = [2, 9]. - b) Clockwise,
i-hat(pointing right) swings down to[0, -1], andj-hat(pointing up) swings to the right to[1, 0]. Matrix[[0, 1], [-1, 0]]. Apply:2 · [0, -1] + 3 · [1, 0] = [0, -2] + [3, 0] = [3, -2]. - c)
i-hatto[1, 0],j-hatto[0, -1]. Matrix[[1, 0], [0, -1]]. Apply:2 · [1, 0] + 3 · [0, -1] = [2, 0] + [0, -3] = [2, -3]. - d) Matrix
[[1, 2], [0, 1]]. Apply:2 · [1, 0] + 3 · [2, 1] = [2, 0] + [6, 3] = [8, 3].
If a matrix looked wrong, check that you wrote the landing spots as columns (top to bottom), not as rows.
Try it yourself, part 2: sketch the unit square’s image
Section titled “Try it yourself, part 2: sketch the unit square’s image”For each matrix, read the two columns (where i-hat and j-hat land), picture the parallelogram they span, and name what the unit square becomes. About 6 minutes.
- a)
[[2, 0], [0, 2]] - b)
[[1, 0], [0, -1]] - c)
[[1, 2], [0, 1]] - d)
[[1, 2], [2, 4]]
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- a)
i-hatto[2, 0],j-hatto[0, 2]: the unit square becomes a 2-by-2 square (uniform scaling, twice as big, same shape). - b)
i-hatto[1, 0],j-hatto[0, -1]: the square flips below the x-axis (a reflection); same size, mirror image. - c)
i-hatto[1, 0],j-hatto[2, 1]: the square leans into a parallelogram tilted to the right (a shear). - d)
i-hatto[1, 2],j-hatto[2, 4]. But[2, 4] = 2 · [1, 2]: the two columns lie on the same line, so the square collapses flat onto a single line segment. The transformation is squashing the whole plane onto a line, a sign you will meet again as a zero determinant.
Flashcards
Section titled “Flashcards”Ten cards. Click any card to reveal the answer. Use the Print flashcards button to lay out the full set as one card per page, ready to print or save as a PDF for offline review.
Q. What two requirements make a transformation linear?
The origin stays fixed (zero maps to zero), and lines stay straight (grid lines stay parallel and evenly spaced, no curving or bunching). Allowed: stretch, rotate, shear, flip. Not allowed: curve or shift the whole plane.
Q. Why does following i-hat and j-hat capture the whole transformation?
Every vector is x · i-hat + y · j-hat, and a linear transformation preserves the combination: L(v) = x · L(i-hat) + y · L(j-hat). Track the two landing spots and you can reconstruct where any vector goes.
Q. What do the columns of a 2x2 matrix represent?
The first column is where i-hat lands; the second column is where j-hat lands. A matrix is just the two transformed basis vectors written side by side.
Q. What is the matrix-vector product M·v in terms of columns?
For v = [x, y], it is x · (first column) + y · (second column). That is the linear combination from the span lesson, reassembling your vector from the transformed basis, not an arbitrary rule.
Q. How do you sketch what a matrix does to the unit square?
Plot the two columns as arrows from the origin, then draw the parallelogram they span. That parallelogram is the image of the unit square and shows the full effect (stretch, rotate, shear, flip, or collapse).
Q. What is the difference between a linear and an affine transformation?
A linear transformation keeps the origin fixed. An affine one can also shift the whole plane by a constant (a translation). A bare 2x2 matrix can only do the linear part; the shift needs an extra added vector.
Q. What is the formula for L(v) in terms of the transformed basis?
L(v) = x · L(i-hat) + y · L(j-hat), where x and y are the coordinates of v. The output uses the same scalars; only the vectors they scale have moved.
Q. What does a shear matrix do to the unit square?
It tips one direction over while leaving the other fixed, so the upright square slants into a leaning parallelogram. Example: [[1, 1], [0, 1]] leaves i-hat put and slides j-hat to [1, 1].
Q. What does it mean if the two columns of a 2x2 matrix are parallel?
The transformation squashes the whole plane onto a single line (the columns span only a line, not the plane). The unit square collapses flat, which you will later recognize as a zero determinant.
Q. Why is matrix-vector multiplication not an arbitrary rule?
Because M · v is defined to be x · col1 + y · col2, the same scale-and-add from the span lesson, applied to the transformed basis vectors. The rule is just bookkeeping for “rebuild the vector from where the basis went.”