Linear algebra and differential equations (Math 54): Lecture 6 - PowerPoint PPT Presentation

Linear subspaces of R 3 Any linear subspace in R 3 is a { 0 } ,

Linear subspaces of R 3 Any linear subspace in R 3 is a { 0 } , a line or plane through the origin,

Linear subspaces of R 3 Any linear subspace in R 3 is a { 0 } , a line or plane through the origin, or all of R 3 .

Linear subspaces of R 3 Any linear subspace in R 3 is a { 0 } , a line or plane through the origin, or all of R 3 . Indeed, any nonzero vector spans a line; two non-colinear vectors span a plane, and three non-coplanar vectors span all of R 3 .

Linear spans

Linear spans The linear span of any collection of vectors is a linear subspace.

Linear spans The linear span of any collection of vectors is a linear subspace. Indeed, if v 1 , . . . , v n are the vectors,

Linear spans The linear span of any collection of vectors is a linear subspace. Indeed, if v 1 , . . . , v n are the vectors, and w = � a i v i and x = � b i v i are linear combinations of the v i ,

Linear spans The linear span of any collection of vectors is a linear subspace. Indeed, if v 1 , . . . , v n are the vectors, and w = � a i v i and x = � b i v i are linear combinations of the v i , then any linear combination of w and x ,

Linear spans The linear span of any collection of vectors is a linear subspace. Indeed, if v 1 , . . . , v n are the vectors, and w = � a i v i and x = � b i v i are linear combinations of the v i , then any linear combination of w and x , �� � �� � � c w + d x = c a i v i + d b i x i = ( ca i + db i ) v i is again a linear combination of the v i .

Ranges

Ranges The range of a linear transformation is a linear subspace.

Ranges The range of a linear transformation is a linear subspace. If v and v ′ are in the range of a transformation T ,

Ranges The range of a linear transformation is a linear subspace. If v and v ′ are in the range of a transformation T , that means there are some w and w ′ with T ( w ) = v and T ( w ′ ) = v ′ .

Ranges The range of a linear transformation is a linear subspace. If v and v ′ are in the range of a transformation T , that means there are some w and w ′ with T ( w ) = v and T ( w ′ ) = v ′ . So, any linear combination

Ranges The range of a linear transformation is a linear subspace. If v and v ′ are in the range of a transformation T , that means there are some w and w ′ with T ( w ) = v and T ( w ′ ) = v ′ . So, any linear combination c v + d v ′ = cT ( w ) + dT ( w ′ ) T ( c w + d w ′ ) =

Ranges The range of a linear transformation is a linear subspace. If v and v ′ are in the range of a transformation T , that means there are some w and w ′ with T ( w ) = v and T ( w ′ ) = v ′ . So, any linear combination c v + d v ′ = cT ( w ) + dT ( w ′ ) T ( c w + d w ′ ) = is in the range of T .

Column space If A is the matrix of the linear transformation T ,

Column space If A is the matrix of the linear transformation T , the range of T is the span of the columns of A .

Column space If A is the matrix of the linear transformation T , the range of T is the span of the columns of A . We call this the column space of A . Indeed, these columns are T ( e 1 ) , · · · , T ( e n ),

Column space If A is the matrix of the linear transformation T , the range of T is the span of the columns of A . We call this the column space of A . Indeed, these columns are T ( e 1 ) , · · · , T ( e n ), so for any v in the range,

Column space If A is the matrix of the linear transformation T , the range of T is the span of the columns of A . We call this the column space of A . Indeed, these columns are T ( e 1 ) , · · · , T ( e n ), so for any v in the range, i.e. v = T ( w ),

Column space If A is the matrix of the linear transformation T , the range of T is the span of the columns of A . We call this the column space of A . Indeed, these columns are T ( e 1 ) , · · · , T ( e n ), so for any v in the range, i.e. v = T ( w ), we can write � w = c i e i hence �� � � v = T ( w ) = T c i e i = c i T ( e i )

Column space Example In a reduced echelon matrix,  0 1 2 0 2 0  0 0 0 1 1 0     0 0 0 0 0 1     0 0 0 0 0 0   0 0 0 0 0 0

Column space Example In a reduced echelon matrix,  0 1 2 0 2 0  0 0 0 1 1 0     0 0 0 0 0 1     0 0 0 0 0 0   0 0 0 0 0 0 the column space

Column space Example In a reduced echelon matrix,  0 1 2 0 2 0  0 0 0 1 1 0     0 0 0 0 0 1     0 0 0 0 0 0   0 0 0 0 0 0 the column space is the set of vectors with zeroes in non-pivot rows.

Column space Example In a reduced echelon matrix,  0 1 2 0 2 0  0 0 0 1 1 0     0 0 0 0 0 1     0 0 0 0 0 0   0 0 0 0 0 0 the column space is the set of vectors with zeroes in non-pivot rows. It’s spanned by the pivot columns.

Column space Example In a reduced echelon matrix,  0 1 2 0 2 0  0 0 0 1 1 0     0 0 0 0 0 1     0 0 0 0 0 0   0 0 0 0 0 0 the column space is the set of vectors with zeroes in non-pivot rows. It’s spanned by the pivot columns. The same is true for an echelon matrix.

Null space