VibeMath

Statistics and linear algebra

Matrix transformations in the coordinate plane

A two-by-two matrix describes where the horizontal and vertical basis vectors move. Multiplying a vector combines those transformed directions, turning matrix arithmetic into a visible motion of the coordinate plane.

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VibeMath Editorial TeamOriginal AI-assisted lesson content edited for clarity, accessibility, and product-truth accuracy.
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Last reviewed

What you will learn

  • Read the columns of a two-by-two matrix as images of the standard basis vectors.
  • Compute a matrix-vector product as a weighted combination of those columns.
  • Verify a ninety-degree rotation by checking direction and preserved vector length.

Build the idea

The standard basis vectors point one unit right and one unit up. A linear transformation is determined by where those two directions go, because every vector x comma y is x copies of the first basis vector plus y copies of the second. A matrix stores the two destination vectors as its columns.

Matrix-vector multiplication rebuilds the input from transformed basis directions. Multiply the first matrix column by x, multiply the second column by y, and add. This column view explains the row calculation and shows why a linear transformation maps lines to lines or, in singular cases, to points; parallel directions remain parallel when they are not collapsed.

For a counterclockwise ninety-degree rotation, the right-pointing basis vector moves up, while the up-pointing basis vector moves left. The corresponding columns are zero comma one and negative one comma zero. Any vector is carried along by that same rotation without changing its length.

A coordinate grid rotating around the origin

Draw the original right basis arrow in one color and up basis arrow in another. Animate the right arrow rotating to point up and the up arrow rotating to point left. Then move a vector from three comma one to negative one comma three.

Visual description: A ninety-degree counterclockwise matrix maps the basis vector one comma zero to zero comma one, maps zero comma one to negative one comma zero, and maps three comma one to negative one comma three.
Read as: R is the two by two matrix with first row zero, negative one, and second row one, zero.
Read as: R times the vector x y equals x times the vector zero one plus y times the vector negative one zero.

Worked example

Problem

Apply the ninety-degree counterclockwise rotation matrix to the vector three comma one, then verify the new vector's direction and length.

Strategy

Scale the transformed basis columns by the input coordinates, add the resulting vectors, and compare squared lengths before and after.

  1. Step 1

    Place the input coordinates into the column vector. The first coordinate weights the transformed right direction, and the second weights the transformed up direction.

    Read as: The matrix zero negative one, one zero, times the vector three one.

  2. Step 2

    Take three copies of the first column and one copy of the second. Adding those arrows gives one unit left and three units up.

    Read as: Three times vector zero one plus one times vector negative one zero equals vector negative one three.

  3. Step 3

    The endpoint moves from quadrant one at three comma one to quadrant two at negative one comma three, matching a counterclockwise quarter turn.

    Read as: R applied to three comma one equals negative one comma three.

  4. Step 4

    Compare squared lengths to avoid unnecessary square roots. Both vectors have squared length ten, so the transformation preserves distance from the origin as a rotation should.

    Read as: Three squared plus one squared equals the quantity negative one, squared, plus three squared, which equals ten.

Answer and verification

The rotated vector is negative one comma three.

A ninety-degree counterclockwise turn maps the direction mostly right and slightly up to a direction slightly left and mostly up. The equal squared lengths confirm that the vector was rotated rather than stretched.

Common mistakes

  • Reading the matrix rows as the transformed basis vectors and predicting the wrong rotation direction.

    For column-vector multiplication, the matrix columns are the images of the standard basis vectors. Test the matrix on one comma zero and zero comma one before transforming a general vector.

  • Multiplying corresponding matrix and vector entries without adding across each output component.

    Either compute row dot products or form a weighted sum of columns. Both methods combine two contributions for each output vector rather than producing four unrelated products.

Try it yourself

Apply the same rotation matrix to the vector negative two comma four and verify that the squared length is preserved.

Show hint

A counterclockwise quarter turn maps x comma y to negative y comma x.

Show answer

The image is negative four comma negative two, and both squared lengths equal twenty.

Accessible lesson transcript

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

  1. Part 1

    Store two transformed arrows

    Begin with the standard basis: one arrow points right and one points up. Every plane vector is a combination of those two arrows. A two-by-two matrix records where the right arrow moves in its first column and where the up arrow moves in its second column. Those two choices determine the whole linear transformation.

  2. Part 2

    Encode a ninety-degree rotation

    Under a counterclockwise quarter turn, the right basis arrow points up, giving column zero comma one. The up basis arrow points left, giving column negative one comma zero. Placing those columns side by side creates the rotation matrix. Testing the two basis inputs confirms the direction before any longer calculation.

  3. Part 3

    Combine the rotated directions

    The vector three comma one contains three copies of the right basis direction and one copy of the up direction. After transformation, take three copies of zero comma one and one copy of negative one comma zero. Adding gives negative one comma three, so the endpoint turns from quadrant one into quadrant two.

  4. Part 4

    Check what stayed unchanged

    A pure rotation should preserve distance from the origin. The original squared length is three squared plus one squared, or ten. The new squared length is the square of negative one plus three squared, also ten. The direction changed by a quarter turn while the length stayed fixed, matching the intended geometry.

Scope and limitations

  • This lesson covers one two-dimensional linear rotation about the origin and does not address translations, affine coordinates, eigenvectors, or changes of basis.
  • Preserving length is special to orthogonal transformations; a general two-by-two matrix may stretch, shear, reflect, collapse, or reverse orientation.