How can a matrix represent a transformation of a vector?

Think of the matrix as a rule that takes an input vector and returns an output vector. With a 2D column vector, each row tells you how to calculate one output coordinate: multiply the row’s entries by the input coordinates, then add.

For example, let A = [[1, 2], [3, 4]] and v = (5, 6). Then

A v = (1·5 + 2·6, 3·5 + 4·6) = (17, 39).

So this transformation sends the input coordinates (5, 6) to the output coordinates (17, 39). Each output coordinate is a row’s dot product with the input. [S1] The calculation describes coordinates in the chosen basis; changing the basis can change the matrix used to describe the same transformation. [S1]

Yes. Keep the same input, (5, 6), and separate the calculation by rows:

1. First row, (1, 2): 1·5 + 2·6 = 17. This becomes the first output coordinate.
2. Second row, (3, 4): 3·5 + 4·6 = 39. This becomes the second output coordinate.

The output is therefore (17, 39). A useful way to check yourself is to match each row with exactly one output coordinate, in order. The first row does not produce the whole vector; it produces its first component. The second row produces its second component. This row-by-row method works for other matrix-vector examples when the dimensions are compatible.

The columns show where the standard basis vectors go. For A = [[1, 2], [3, 4]], the first column is (1, 3), and the second is (2, 4). Under this matrix transformation, the basis vectors (1, 0) and (0, 1) map to those respective columns. [S1]

Your input (5, 6) can be written as 5(1, 0) + 6(0, 1). Its transformed vector is consequently 5(1, 3) + 6(2, 4) = (17, 39). [S1] That column-based view and the row-by-row calculation describe the same multiplication in two ways. Geometrically, applying a matrix can move the corners of the unit square into a parallelogram; the matrix columns give the transformed directions of its sides. [S2] These explanations use the stated column-vector convention; another convention for writing vectors may require different notation. [S2]

Use this sequence:

1. Write the input as a column vector and copy the matrix clearly.
2. Take one matrix row at a time; multiply corresponding entries and add.
3. Put each row’s sum in the matching position of the output vector.
4. Check that you used every row once and kept the output coordinates in order.
5. For a geometric sense-check, compare the matrix columns with the images of the two standard basis vectors. [S1]

For the worked example, the reusable layout is:

[[1, 2], [3, 4]] · (5, 6) = (1·5 + 2·6, 3·5 + 4·6) = (17, 39)

This gives the coordinate calculation and its result, not a sketch of the transformed vector. The sources establish the row-dot-product calculation and the basis-column interpretation for the stated conventions; they do not determine a particular application’s coordinate system or how to draw a graph to scale. [S1][S2]