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Homework 5 (due Wednesday 8 th Nov midnight) Use this definition for Column Space of a Matrix Column Space of a matrix A is the set ColA of all linear combinations of the columns of A. In other words, if A = [a 1 , a 2 ,…..,a n ] with columns in R m , then ColA is the same as span{ a 1 , a 2 ,…..,a n }. 1.Find basis for each of the following subspaces: column space ColA a) b) c) 2. Write a system of equations that is equivalent to the given vector equation. a) b)
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Homework 5 - University of California, Irvinegraphics.ics.uci.edu/ICS6N/Homework/HW5.pdf · 2017. 11. 3. · Homework 5 (due Wednesday 8th Nov midnight) Use this definition for Column

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Page 1: Homework 5 - University of California, Irvinegraphics.ics.uci.edu/ICS6N/Homework/HW5.pdf · 2017. 11. 3. · Homework 5 (due Wednesday 8th Nov midnight) Use this definition for Column

Homework 5 (due Wednesday 8th Nov midnight)

Use this definition for Column Space of a Matrix

Column Space of a matrix A is the set ColA of all linear combinations of the columns of A. In other words, if A = [a1, a2,…..,an] with columns in Rm, then ColA is the same as span{ a1, a2,…..,an}.

1.Find basis for each of the following subspaces: column space ColA

a)

b)

c)

2. Write a system of equations that is equivalent to the given vector equation.

a)

b)

Page 2: Homework 5 - University of California, Irvinegraphics.ics.uci.edu/ICS6N/Homework/HW5.pdf · 2017. 11. 3. · Homework 5 (due Wednesday 8th Nov midnight) Use this definition for Column

3. Write a vector equation that is equivalent to the given system of equations.

a)

b)

4. Check if the column vectors of matrix A are linearly independent or not.

a)

b)

5. Write the following vector equations as matrix equations.

a)

b)

Page 3: Homework 5 - University of California, Irvinegraphics.ics.uci.edu/ICS6N/Homework/HW5.pdf · 2017. 11. 3. · Homework 5 (due Wednesday 8th Nov midnight) Use this definition for Column

6. Write the augmented system for the linear system that corresponds to the matrix equation Ax = b. Solve the system and write the solution as a vector.

a)

b)

7. Describe all solution of Ax=0 in parametric vector form, where A is row-equivalent to the given matrix.

a)

b)

8. Let H be the set of all vectors of the form−2𝑡5𝑡3𝑡

. Find a vector v in ℝ( such that𝐻 = 𝑆𝑝𝑎𝑛{𝑣}. Why does

this show that H is a subspace of ℝ(.

9. Let W be the set of all vectors of the form2𝑏 + 3𝑐−𝑏2𝑐

. Where b and c are arbitrary. Find vectors u and v such

that 𝑊 = 𝑆𝑝𝑎𝑛{𝑣, 𝑢}. Why does this show that W is a subspace of ℝ(.

10. Given subspaces H and K of a vector space V, the sum of H and K, written as 𝐻 + 𝐾, is the set of all vectors in V that can be written as the sum of two vectors, one in H and the other one in K, that is,

𝐻 + 𝐾 = {𝑤:𝑤 = 𝑢 + 𝑣𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑢𝑖𝑛𝐻𝑎𝑛𝑑𝑠𝑜𝑚𝑒𝑣𝑖𝑛𝐾}

a. Show that H+K is a subspace of V. b. Show that H is a subspace of H+K and K is a subspace of H+K.

Page 4: Homework 5 - University of California, Irvinegraphics.ics.uci.edu/ICS6N/Homework/HW5.pdf · 2017. 11. 3. · Homework 5 (due Wednesday 8th Nov midnight) Use this definition for Column

11. Mark each statement True or False. Justify each answer.

a. A subset H of a vector space V is a subspace of V if the zero vector is in H.

b. A subspace is also a vector space.

c. A vector space is also a subspace.

d. ℝC is a subspace of ℝ(.

12. Let W = span{v1, …. vp}. Show that if x is orthogonal to each vj, for 1 ≤ j ≤ p, then x is orthogonal to every vector in W.

13. Let 𝑈 =

12

23

12

−2 30 1

3

𝑎𝑛𝑑𝑥 = 23.Notice that U had orthonormal columns and

14.

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Page 5: Homework 5 - University of California, Irvinegraphics.ics.uci.edu/ICS6N/Homework/HW5.pdf · 2017. 11. 3. · Homework 5 (due Wednesday 8th Nov midnight) Use this definition for Column

16. Mark each statement as True or False and justify your answer.

In the exercise 17-22 determine which sets of vectors are orthonormal. If a set is only orthogonal, normalize the vectors to produce orthonormal sets.

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Page 6: Homework 5 - University of California, Irvinegraphics.ics.uci.edu/ICS6N/Homework/HW5.pdf · 2017. 11. 3. · Homework 5 (due Wednesday 8th Nov midnight) Use this definition for Column

24.

a. Let U = [u1, u2]. Compute UTU and UUT. b. Compute projW y and (UUT)y.

25.

26. The given set is basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.

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