6.2 Orthogonal complements and the matrix transpose

Kristen Pueschel

6.2 Orthogonal complements and the matrix transpose

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Try out the Preview Activity and read Orthogonal complements and the matrix transpose in Understanding Linear Algebra by David Austin.

Definition: Given a subspace

H

of

R^{n}

, the orthogonal complement of

H

is the set of vectors in

R^{n}

, each of which is orthogonal to every vector in

H

. We denote the orthogonal complement by

H^{⊥}

.

Definition: Consider an

m \times n

matrix

A

with entries

a_{i j}

. The transpose of $A$ , written

A^{T}

, is the

n \times m

matrix with entries

[A^{T}]_{j i} = a_{i j}

. In other words, the rows of

A

are columns of

A^{T}

and the columns of

A

are rows of

A^{T}

.

If the matrix A has columns ${\vec{v}}_{1}, \dots, {\vec{v}}_{n}$ , then $A^{T} \vec{x} = [(\begin{matrix} {\vec{v}}_{1} \cdot \vec{x} \\ ⋮ \\ {\vec{v}}_{n} \cdot \vec{x} \end{matrix})$ . We see that we can compute all n dot products by finding this product.

$N u l (A^{T}) = (C o l (A))^{⊥}$ .

Proof: $A^{T} \vec{x} = \vec{0}$ if and only if $\vec{x}$ is orthogonal to every column of $A$ . The columns of $A$ span $C o l (A)$ , so $\vec{x}$ is orthogonal to all element of $C o l (A)$ .

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