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$A 2\times 2$ transformation matrix maps the standard basis vectors to its own columns. Specifically, the first column of the matrix is the image of the vector $[1, 0]^T$, and the second column is the image of the vector $[0, 1]^T$.

Conditions: The matrix is $2\times 2$.; Working in standard Cartesian coordinates.

$A 2\times 2$ transformation matrix maps the standard basis vectors to its own columns. Specifically, the first column of the matrix is the image of the vector $[1, 0]^T$, and the second column is the image of the vector $[0, 1]^T$.

Conditions: The matrix is $2\times 2$.; Working in standard Cartesian coordinates.

The mathematician generalizes the concept to any object that supports sensible addition and scalar multiplication operations.

Conditions: Abstract linear algebra context

The mathematician generalizes the concept to any object that supports sensible addition and scalar multiplication operations.

Conditions: Abstract linear algebra context

Multiplying a 3x3 matrix by a vector $(x, y, z)$ scales each column of the matrix (which represents a transformed basis vector) by the corresponding input coordinate and sums the results. This works because linear transformations preserve addition and scalar multiplication.

Conditions: Matrix is 3x3 representing a linear transformation; Input vector has coordinates $(x, y, z)$; Transformation is linear

Multiplying a 3x3 matrix by a vector $(x, y, z)$ scales each column of the matrix (which represents a transformed basis vector) by the corresponding input coordinate and sums the results. This works because linear transformations preserve addition and scalar multiplication.

Conditions: Matrix is 3x3 representing a linear transformation; Input vector has coordinates $(x, y, z)$; Transformation is linear

Matrix-vector multiplication in 3D works by scaling the columns of the matrix by the corresponding coordinates of the input vector and summing the results. The coordinates $(x, y, z)$ act as scalar multipliers for the transformed basis vectors (the columns), leveraging the linearity property that preserves addition and scalar multiplication.

Conditions: The matrix is a 3x3 transformation matrix.; The input vector has coordinates $(x, y, z)$.; The transformation is linear.

Matrix-vector multiplication in 3D works by scaling the columns of the matrix by the corresponding coordinates of the input vector and summing the results. The coordinates $(x, y, z)$ act as scalar multipliers for the transformed basis vectors (the columns), leveraging the linearity property that preserves addition and scalar multiplication.

Conditions: The matrix is a 3x3 transformation matrix.; The input vector has coordinates $(x, y, z)$.; The transformation is linear.

Matrix-vector multiplication is reframed as scaling the transformed basis vectors (which are the columns of the matrix) by the input vector's components and summing them. This constructs the final position within the skewed coordinate system defined by the matrix, rather than just following a tedious arithmetic recipe.

Conditions: Input and output coordinates use the fixed standard basis.; The matrix represents an active linear map moving vectors.

Matrix-vector multiplication is reframed as scaling the transformed basis vectors (which are the columns of the matrix) by the input vector's components and summing them. This constructs the final position within the skewed coordinate system defined by the matrix, rather than just following a tedious arithmetic recipe.

Conditions: Input and output coordinates use the fixed standard basis.; The matrix represents an active linear map moving vectors.

The residual vector $\vec{r} = A\vec{x}^* - \vec{b}$ is orthogonal to the column space $C(A)$ because $A\vec{x}^*$ is the orthogonal projection of $\vec{b}$ onto $C(A)$. Orthogonality to $C(A)$ means $\vec{r}$ is in the orthogonal complement $C(A)^\perp$.

Conditions: $A\vec{x}^*$ is the orthogonal projection of $\vec{b}$ onto $C(A)$; $C(A)^\perp = N(A^T)$ (Fundamental Theorem of Linear Algebra); Matrix multiplication distributes over subtraction

The residual vector $\vec{r} = A\vec{x}^* - \vec{b}$ is orthogonal to the column space $C(A)$ because $A\vec{x}^*$ is the orthogonal projection of $\vec{b}$ onto $C(A)$. Orthogonality to $C(A)$ means $\vec{r}$ is in the orthogonal complement $C(A)^\perp$.

Conditions: $A\vec{x}^*$ is the orthogonal projection of $\vec{b}$ onto $C(A)$; $C(A)^\perp = N(A^T)$ (Fundamental Theorem of Linear Algebra); Matrix multiplication distributes over subtraction