MA 265 REVIEW FOR MIDTERM #2
GOINS
4: Real Vector Spaces
4.1: Vectors in the Plane and 3-Space A vector in the plane or a 2-vector is a 2 1 matrix x = x y . The real numbers x and y are called
the components of x. Given two ordered pairs P (x, y ) and Q(x
MA 265 GOINS SAMPLE MIDTERM EXAMINATION #2
Name:
Instructions:
Circle the correct answer on the following pages. You have 50 minutes to complete 25 problems. No textbooks, personal notes, calculators, or computing aids are allowed during the examination p
MA 265 GOINS MIDTERM EXAMINATION #2
Name:
Instructions:
Circle the correct answer on the following pages. You have 50 minutes to complete 25 problems. No textbooks, personal notes, calculators, or computing aids are allowed during the examination period.
MA 265 GOINS MIDTERM EXAMINATION #1
Name:
Instructions:
Circle the correct answer on the following pages. You have 50 minutes to complete 35 problems. No textbooks, personal notes, calculators, or computing aids are allowed during the examination period.
MA 265 LECTURE NOTES: WEDNESDAY, APRIL 23
Systems with Diagonalizable Matrices Example. Consider the rst-order homogeneous system dx = x(t) y (t) dt dy = 2 x(t) + 4 y (t) dt We nd the general solution. The idea is to make a change of variables to make thi
MA 265 LECTURE NOTES: MONDAY, APRIL 21
Differential Equations Dierences vs. Dierentials. Consider a function x = x(t). Given two points P = (t0 , x0 ) and Q = (t1 , x1 ) on the graph of this function, we can draw a secant line through them. The slope of t
MA 265 LECTURE NOTES: FRIDAY, APRIL 18
Review of Complex Numbers Denitions. Consider the quadratic equation a z 2 + b z + c = 0 where a, b, and c are real numbers. Formally, we can write the two roots of this equation as b2 4 c a b z= . 2a 2a Recall that
MA 265 LECTURE NOTES: WEDNESDAY, APRIL 16
Matrices That Are Diagonalizable Distinct Eigenvalues. Let A be an n n matrix. We show that if A has distinct eigenvalues, then A can be diagonalized. To see why, let cfw_1 , 2 , . . . , n denote the distinct eig
MA 265 LECTURE NOTES: MONDAY, APRIL 14
Similar Matrices Review. Let V = R , and say that L : V V is a linear operator. We have seen that there is an n n matrix A such that L(u) = A u; this is the standard matrix representing L. Similarly, we have seen tha
MA 265 LECTURE NOTES: FRIDAY, APRIL 11
Eigenvalues and Eigenvectors Review. Let (V, +, ) be a real vector space of dimension n. Recall that a linear operator is a function L : V V such that L preserves vector addition: For all u, v V , we have L (u + v) =
MA 265 LECTURE NOTES: WEDNESDAY, APRIL 9
Least Squares Best Polynomial Approximation. We explain how to use the concept of a least squares solution to nd a polynomial t to a collection of data. Say that we have a list of data as follows: Time t1 t2 . . .
MA 265 LECTURE NOTES: MONDAY, APRIL 7
Orthogonal Complements Example. Consider the 3 3 matrix 1 1 A= 1 2 2 1 0 2 1 4 1 3 . 3 2 1 1 3 5 1 5
We compute (1) the orthogonal complement for the row space of A and (2) the orthogonal complement for the column spa
MA 265 LECTURE NOTES: WEDNESDAY, APRIL 2
Orthogonal Complements Properties. Let V , +, , (, ) be an inner product space. Let W is a subspace of V , and W be its orthogonal complement: W = vV (v, w) = 0 for all w W .
The following properties hold. (1) W is
MA 265 LECTURE NOTES: MONDAY, MARCH 31
Gram-Schmidt Process Orthogonalization. Let V , +, , (, ) be an inner product space. Say that T = cfw_v1 , v2 , . . . , vn is a collection of of nonzero vectors, and let W = span T be their span. Last time, we showe
MA 265 LECTURE NOTES: FRIDAY, MARCH 28
Inner Product Spaces Length and Direction in Inner Product Spaces. Let V , +, , (, ) be an inner product space. We dene length, distance, and direction in the following way: The length of a vector v V is given by |v|
MA 265 LECTURE NOTES: WEDNESDAY, MARCH 26
Inner Product Spaces Direction in R2 and R3 . We have seen that the angle between two vectors u and v in R2 is given by u1 v1 + u2 v2 = arccos where 0 . |u| |v| Similarly, the angle between two vectors u and v in
MA 265 LECTURE NOTES: MONDAY, MARCH 24
Inner Product Spaces Length and Direction in R2 . Recall that V = R2 is the collection of all 2-vectors: R2 = v= v1 v2 v1 , v 2 R .
We can dene the length of a vector v by drawing a triangle with base v1 , height v2
MA 265 LECTURE NOTES: FRIDAY, MARCH 21
Row and Column Space Rank of a Matrix. Let A be an m n matrix. We have seen that the row rank of A is the dimension of the subspace of Rn spanned by the rows of A; and that the column space of A is the dimension of t
MA 265 LECTURE NOTES: WEDNESDAY, MARCH 19
Homogeneous Systems Relationship with Nonhomogeneous Systems. In general, consider a nonhomogeneous system A x = b. Recall that the set of solutions does not form a linear space: x Rn A x = b . Say that xp is a pa
MA 265 LECTURE NOTES: MONDAY, MARCH 17
Homogeneous Systems Example. We recall ideas from the previous lecture. We considered the following homogeneous system of linear equations x1 + x2 + 4 x3 + x4 + 2 x5 = 0 x2 + 2 x3 + x4 + x5 = 0 x4 + 2 x5 = 0 x1 x2 +
MA 265 LECTURE NOTES: FRIDAY, MARCH 7
Bases and Dimension Basic Results. Let (V, +, ) be a real vector space of dimension n. We recall a few statements we made without proof at the end of the previous lecture. (1) (2) (3) (4) (5) (6) If T is a maximal ind
MA 265 LECTURE NOTES: WEDNESDAY, MARCH 5
Bases and Dimension Dimension. Let (V, +, ) be a real vector space. Let W = span S be a subspace of V in terms of a linearly independent set S = cfw_v1 , v2 , . . . , vn . If T = cfw_w1 , w2 , . . . , wm is any ba
MA 265 LECTURE NOTES: MONDAY, MARCH 3
Bases and Dimension Bases and Span. Let (V, +, ) be a real vector space. Let W = span S be the span of some collection S = cfw_v1 , v2 , . . . , vk of vectors from V . Then W has a basis T S . We sketch the proof via
MA 265 LECTURE NOTES: FRIDAY, FEBRUARY 29
Linear Independence (contd) Linear Independence and Span. Let (V, +, ) be a real vector space. Let S = cfw_v1 , v2 , . . . , vk be a collection of vectors from V . Recall that W = span S is a subspace of V , and
MA 265 LECTURE NOTES: WEDNESDAY, FEBRUARY 27
Linear Independence (contd) Example. Let V = M22 be the real vector space corresponding to the 2 2 matrices. Consider the following three vectors: 21 12 0 3 v1 = , v2 = , and v3 = . 01 10 2 1 We determine wheth
MA 265 LECTURE NOTES: MONDAY, FEBRUARY 25
Span (contd) Example. We nish the example we began in the previous lecture. Consider the homogeneous linear system A x = 0 in terms of the 4 4 matrix 1 1 0 2 2 2 1 5 . A= 1 1 1 3 4 4 1 9 Recall that the null space
MA 265 LECTURE NOTES: FRIDAY, FEBRUARY 22
Span Linear Combinations. Let (V, +, ) be a real vector space. Say that S = cfw_v1 , v2 , . . . , vk is a subset of vectors from V . Recall that a vector v V is called a linear combination if there exist scalars
MA 265 LECTURE NOTES: WEDNESDAY, FEBRUARY 20
Subspaces Denition. Let (V, , ) be a real vector space. We say that a subset W V is a subspace of V if (1) W = , and (2) (W, , ) is a real vector space. We spend the rest of the lecture giving examples and disc
MA 265 LECTURE NOTES: MONDAY, FEBRUARY 18
Real Vector Spaces Properties of n-Space. In the previous lecture, we studied properties of V= R2 R3 as 2-space; and as 3-space.
If u, v, w V and c, d R then the following properties are valid: (Commutativity: ) u
MA 265 LECTURE NOTES: WEDNESDAY, FEBRUARY 13
Real Vector Spaces Vectors in n-Space. Recall that Rn is the collection of all real n-vectors: x1 x2 x= . . . xn where the entries xi are real numbers. We call this collection (real) n-space. The elements x Rn