# Vector Space

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### r : the row rank of A

= n A m r B Cn r m

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### ×n

the row space of A is the span of the rows of C

the column space of A is the span of the columns of B

### r : the row rank of A

= n A m r B Cn r m

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### ×n

the row space of A is the span of the rows of C

the column space of A is the span of the columns of B

a11 a12 a21 a22 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1x b2 x b1y b2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ c1x c1y c2 x c2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1xc1x + b2 xc2 x b1xc1y + b2 xc2 y b1yc1x + b2 yc2 x b1yc1y + b2 yc2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ a11 a12 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1xc1x + b2 xc2 x b1xc1y + b2 xc2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1x c1x c1y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥+ b2 x c2 x c2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥

⎥: the 1st row vector a21 a22 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1yc1x + b2 yc2 x b1yc1y + b2 yc2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1y c1x c1y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥+ b2 y c2 x c2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥

⎥: the 2nd row vector

### r : the row rank of A

= n A m r B Cn r m

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### ×n

the row space of A is the span of the rows of C

the column space of A is the span of the columns of B

a11 a12 a21 a22 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1x b2 x b1y b2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ c1x c1y c2 x c2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1xc1x + b2 xc2 x b1xc1y + b2 xc2 y b1yc1x + b2 yc2 x b1yc1y + b2 yc2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ a11 a21 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1xc1x + b2 xc2 x b1yc1x + b2 yc2 x ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = c1x b1x b1y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥+ c2 x b2 x b2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥

⎥: the 1st column vector a12 a22 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1xc1y + b2 xc2 y b1yc1y + b2 yc2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = c1y b1x b1y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥+ c2 y b2 x b2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥

⎥: the 2nd column vector

### r : the row rank of A

= n A m r B Cn r m

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### ×n

=

the row space of A is the span of the rows of C

the column space of A is the span of the columns of B

### r : the row rank of A

m n A m r B Cn r

the row rank of A = the number of independent row vectors of C = r

the column rank of A = the number of independent column vectors of B = r

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Row Space

: the subspace of !n spanned by the row vectors of A

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Column Space

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### Null Space

Null Space

: the set of vectors x such that Ax=0.

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Figure

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