Practice Problems - Columbia Math - Columbiabayer/LinearAlgebra/S12/Practice2-Solutions... ·...
Transcript of Practice Problems - Columbia Math - Columbiabayer/LinearAlgebra/S12/Practice2-Solutions... ·...
![Page 1: Practice Problems - Columbia Math - Columbiabayer/LinearAlgebra/S12/Practice2-Solutions... · PracticeProblems2 Linear Algebra, Dave Bayer, March 18, 2012 [1] Let V and W be the subspaces](https://reader031.fdocuments.net/reader031/viewer/2022030406/5a7ee6aa7f8b9a4d628ee19b/html5/thumbnails/1.jpg)
Practice Problems Linear Algebra, Dave Bayer, March ,
[] Let V and W be the subspaces of R2 spanned by (1,1) and (1,2), respectively. Find vectors v ∈ V
and w ∈ W so v+w = (2,−1).
![Page 2: Practice Problems - Columbia Math - Columbiabayer/LinearAlgebra/S12/Practice2-Solutions... · PracticeProblems2 Linear Algebra, Dave Bayer, March 18, 2012 [1] Let V and W be the subspaces](https://reader031.fdocuments.net/reader031/viewer/2022030406/5a7ee6aa7f8b9a4d628ee19b/html5/thumbnails/2.jpg)
[] Let V andW be the subspaces of R2 spanned by (1,−1) and (2,1), respectively. Find vectors v ∈ V
and w ∈ W so v+w = (1,1).
![Page 3: Practice Problems - Columbia Math - Columbiabayer/LinearAlgebra/S12/Practice2-Solutions... · PracticeProblems2 Linear Algebra, Dave Bayer, March 18, 2012 [1] Let V and W be the subspaces](https://reader031.fdocuments.net/reader031/viewer/2022030406/5a7ee6aa7f8b9a4d628ee19b/html5/thumbnails/3.jpg)
[] Let V and W be the subspaces of R2 spanned by (1,1) and (1,4), respectively. Find vectors v ∈ V
and w ∈ W so v+w = (2,3).
![Page 4: Practice Problems - Columbia Math - Columbiabayer/LinearAlgebra/S12/Practice2-Solutions... · PracticeProblems2 Linear Algebra, Dave Bayer, March 18, 2012 [1] Let V and W be the subspaces](https://reader031.fdocuments.net/reader031/viewer/2022030406/5a7ee6aa7f8b9a4d628ee19b/html5/thumbnails/4.jpg)
[] Let V be the subspace of R3 consisting of all solutions to the equation x+ y+ z = 0. LetW be thesubspace of R3 spanned by (1,1,0). Find vectors v ∈ V and w ∈ W so v+w = (0,0,1).
![Page 5: Practice Problems - Columbia Math - Columbiabayer/LinearAlgebra/S12/Practice2-Solutions... · PracticeProblems2 Linear Algebra, Dave Bayer, March 18, 2012 [1] Let V and W be the subspaces](https://reader031.fdocuments.net/reader031/viewer/2022030406/5a7ee6aa7f8b9a4d628ee19b/html5/thumbnails/5.jpg)
[] Let V be the subspace of R3 consisting of all solutions to the equation x+ y− z = 0. LetW be thesubspace of R3 spanned by (1,0,4). Find vectors v ∈ V and w ∈ W so v+w = (1,1,1).
![Page 6: Practice Problems - Columbia Math - Columbiabayer/LinearAlgebra/S12/Practice2-Solutions... · PracticeProblems2 Linear Algebra, Dave Bayer, March 18, 2012 [1] Let V and W be the subspaces](https://reader031.fdocuments.net/reader031/viewer/2022030406/5a7ee6aa7f8b9a4d628ee19b/html5/thumbnails/6.jpg)
[] Let V be the subspace of R3 consisting of all solutions to the equation x+ 2y+ z = 0. LetW be thesubspace of R3 spanned by (1,1,1). Find vectors v ∈ V and w ∈ W so v+w = (1,1,0).
![Page 7: Practice Problems - Columbia Math - Columbiabayer/LinearAlgebra/S12/Practice2-Solutions... · PracticeProblems2 Linear Algebra, Dave Bayer, March 18, 2012 [1] Let V and W be the subspaces](https://reader031.fdocuments.net/reader031/viewer/2022030406/5a7ee6aa7f8b9a4d628ee19b/html5/thumbnails/7.jpg)
[] Let V be the subspace of R4 consisting of all solutions to the system of equations
[1 1 0 00 0 1 1
]w
x
y
z
=
[00
].
LetW be the orthogonal complement of V . Find vectors v ∈ V and w ∈ W so v+w = (1,0,1,0).
![Page 8: Practice Problems - Columbia Math - Columbiabayer/LinearAlgebra/S12/Practice2-Solutions... · PracticeProblems2 Linear Algebra, Dave Bayer, March 18, 2012 [1] Let V and W be the subspaces](https://reader031.fdocuments.net/reader031/viewer/2022030406/5a7ee6aa7f8b9a4d628ee19b/html5/thumbnails/8.jpg)
[] Let V be the subspace of R4 consisting of all solutions to the system of equations
[1 1 1 10 1 1 1
]w
x
y
z
=
[00
].
LetW be the orthogonal complement of V . Find vectors v ∈ V and w ∈ W so v+w = (0,0,1,1).
![Page 9: Practice Problems - Columbia Math - Columbiabayer/LinearAlgebra/S12/Practice2-Solutions... · PracticeProblems2 Linear Algebra, Dave Bayer, March 18, 2012 [1] Let V and W be the subspaces](https://reader031.fdocuments.net/reader031/viewer/2022030406/5a7ee6aa7f8b9a4d628ee19b/html5/thumbnails/9.jpg)
[] Let V be the subspace of R4 consisting of all solutions to the system of equations
[0 1 2 33 2 1 0
]w
x
y
z
=
[00
].
LetW be the orthogonal complement of V . Find vectors v ∈ V and w ∈ W so v+w = (1,0,0,0).