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← Tensor Foundations step 3 of 7
Transpose a Matrix
A matrix transpose exchanges rows and columns. Input row i becomes output
column i: output[j, i] = x[i, j].
Your task
Implement transpose(x) for a two-dimensional tensor of shape (m, n).
Return shape (n, m), preserving the values, dtype and device. Leave x
unchanged. No batched or higher-dimensional transpose is needed.
For x=[[1, 2, 3], [4, 5, 6]], return [[1, 4], [2, 5], [3, 6]].
The first input row, [1, 2, 3], becomes the first output column.
Use x.T in either framework. PyTorch also provides x.transpose(0, 1);
JAX provides jnp.transpose(x).
Changing the shape alone is not enough. Reshaping the example to (3, 2)
would produce [[1, 2], [3, 4], [5, 6]], which is a different arrangement.
A square matrix keeps its shape when transposed, but its off-diagonal values
still exchange positions. Transposing twice restores the original matrix.
Understand the ideas behind this exercise. The Problem tab contains the exact implementation requirements.
Give each tensor axis a meaning
A tensor's shape tells you how many entries each axis has. It does not tell you
what those entries represent. For example, (batch, features) and
(features, batch) use different axis orders. Name the axes before choosing an
operation.
A reduction combines entries along an axis, for example by summing or
averaging. Averaging the features of a (B, D) tensor gives one number per
example, with shape (B,). Keeping the reduced axis gives shape (B, 1) instead.
For [[2, 4, 6], [1, 3, 5]], the row means with the axis kept are [[4], [3]].
Subtracting them from the input gives [[-2, 0, 2], [-2, 0, 2]]: each row has
been centered using its own mean.
Reshape and transpose do different jobs
A reshape regroups elements using the operation's traversal order. A transpose
changes the axis order. For the usual row-major reshape, [[1, 2, 3], [4, 5, 6]]
reshaped to (3, 2) becomes [[1, 2], [3, 4], [5, 6]]. Transposing the original
instead gives [[1, 4], [2, 5], [3, 6]].
Both results have shape (3, 2), but their entries mean different things. Check
values as well as shapes. Unequal axis sizes make many axis mistakes easier to
spot than square examples do.
Concatenate along an axis or stack on a new one
Concatenation extends an existing axis. Stacking creates a new axis. For two
arrays of shape (2, 3), concatenating on axis zero gives (4, 3). Stacking on
a new leading axis gives (2, 2, 3).
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