homework 5 - university of california, irvinegraphics.ics.uci.edu/ics6n/homework/hw5.pdf · 2017....
TRANSCRIPT
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)
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)
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.
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.
15.
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