4. (a) Let A E Mmxn (R). Let W₁ CR" be the row space of A (i.e. the span of the row vectors of A), and let W₂ C Rn be the solution space of the homogeneous system of linear equations Ax 0. Show that W₁ and W2 are orthogonal complementary pair in R". = (b) Show that any subspace of R" is the solution space of some homogeneous system of linear equations. More precisely, show that if W CR" is a subspace of dimension m, then there exists a homogeneous system of (n - m) linear equations in n unknowns whose solution space is W. (Hint: use (a).)
4. (a) Let A E Mmxn (R). Let W₁ CR" be the row space of A (i.e. the span of the row vectors of A), and let W₂ C Rn be the solution space of the homogeneous system of linear equations Ax 0. Show that W₁ and W2 are orthogonal complementary pair in R". = (b) Show that any subspace of R" is the solution space of some homogeneous system of linear equations. More precisely, show that if W CR" is a subspace of dimension m, then there exists a homogeneous system of (n - m) linear equations in n unknowns whose solution space is W. (Hint: use (a).)
Elementary Linear Algebra (MindTap Course List)
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Chapter4: Vector Spaces
Section4.4: Spanning Sets And Linear Independence
Problem 74E: Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors...
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![4. (a) Let A € Mmxn(R). Let W₁ ≤ R" be the row space of A (i.e. the span of the
row vectors of A), and let W₂ Rn be the solution space of the homogeneous
system of linear equations Ax 0. Show that W₁ and W₂ are orthogonal
complementary pair in R".
=
(b) Show that any subspace of R" is the solution space of some homogeneous
system of linear equations. More precisely, show that if WR" is a subspace
of dimension m, then there exists a homogeneous system of (n - m) linear
equations in n unknowns whose solution space is W. (Hint: use (a).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2e948f6-fd6f-485f-942e-c931230f8579%2Ff17eb9f6-5a7b-418c-9a7d-899a9031b9b8%2Fc4h6rn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. (a) Let A € Mmxn(R). Let W₁ ≤ R" be the row space of A (i.e. the span of the
row vectors of A), and let W₂ Rn be the solution space of the homogeneous
system of linear equations Ax 0. Show that W₁ and W₂ are orthogonal
complementary pair in R".
=
(b) Show that any subspace of R" is the solution space of some homogeneous
system of linear equations. More precisely, show that if WR" is a subspace
of dimension m, then there exists a homogeneous system of (n - m) linear
equations in n unknowns whose solution space is W. (Hint: use (a).)
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