Vector Spaces - UCSB Physics
... corresponding to the vector |xi, we see that |x0 i + |y 0 i = |(x + y)0 i and α|x0 i = |(αx)0 i. The vector |x0 i is called projection of the vector |xi on the subspace spanned by the orthonormal system. By the definition of the span ν̃, corresponding ONS forms an orthonormal basis (ONB) in it, that ...
... corresponding to the vector |xi, we see that |x0 i + |y 0 i = |(x + y)0 i and α|x0 i = |(αx)0 i. The vector |x0 i is called projection of the vector |xi on the subspace spanned by the orthonormal system. By the definition of the span ν̃, corresponding ONS forms an orthonormal basis (ONB) in it, that ...
1. (14 points) Consider the system of differential equations dx1 dt
... (a) Solve the system if k = −2, x1 (0) = −3 and x2 (0) = 3. (b) Sketch the phase portrait for this system when k = −2. (c) For which values of k will the trajectories in the phase portrait be spirals into the origin? spirals out of the origin? Explain. 2. (14 points) Let R be the region in the plane ...
... (a) Solve the system if k = −2, x1 (0) = −3 and x2 (0) = 3. (b) Sketch the phase portrait for this system when k = −2. (c) For which values of k will the trajectories in the phase portrait be spirals into the origin? spirals out of the origin? Explain. 2. (14 points) Let R be the region in the plane ...
Dot Product, Cross Product, Determinants
... Parallelogram in R3 For d = 3 the sum in (4) contains three terms and we have A2 = (a2 b3 − a3 b2 )2 + (a3 b1 − a1 b3 )2 + (a1 b2 − a2 b1 )2 . Hence A = k~xk with the vector ~x defined by ~x := (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ). Note that x j = ak b` − a` bk where 1. j, k, ` are diff ...
... Parallelogram in R3 For d = 3 the sum in (4) contains three terms and we have A2 = (a2 b3 − a3 b2 )2 + (a3 b1 − a1 b3 )2 + (a1 b2 − a2 b1 )2 . Hence A = k~xk with the vector ~x defined by ~x := (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ). Note that x j = ak b` − a` bk where 1. j, k, ` are diff ...
Vectors and Vector Operations
... of all linear combinations of two fixed vectors. In general, a subspace consists of all linear combinations of a fixed set of vectors. However, the standard defintion of a subspace is a little different. Definition 1. A collection S of vectors is a subspace if it has the following two ...
... of all linear combinations of two fixed vectors. In general, a subspace consists of all linear combinations of a fixed set of vectors. However, the standard defintion of a subspace is a little different. Definition 1. A collection S of vectors is a subspace if it has the following two ...
CM222, Linear Algebra Mock Test 3 Solutions 1. Let P2 denote the
... Explanation: a) is the definition. c) is the same as a) since the eigenvalues with eigenvalue λ are precisely the non-zero vectors in the null space of A − λI. b) is incorrect, since for example the geometric multiplicity of the eigenvalue 1 for the 2 × 2 identity matrix is 2, but the minimial polyn ...
... Explanation: a) is the definition. c) is the same as a) since the eigenvalues with eigenvalue λ are precisely the non-zero vectors in the null space of A − λI. b) is incorrect, since for example the geometric multiplicity of the eigenvalue 1 for the 2 × 2 identity matrix is 2, but the minimial polyn ...
Question 1 2 3 4 5 6 7 8 9 10 Total Score
... (z − 7)x3 = 0, which always has a solution, namely x3 = 0. Another common mistake was to assert that the pivot entry must be 1. While pivots are indeed always 1, anything except 0 can be turned into a 1 by row operations. ...
... (z − 7)x3 = 0, which always has a solution, namely x3 = 0. Another common mistake was to assert that the pivot entry must be 1. While pivots are indeed always 1, anything except 0 can be turned into a 1 by row operations. ...
SELECTED SOLUTIONS FROM THE HOMEWORK 1. Solutions 1.2
... 1.3, 24 Explain why a set of n vectors in Rn is linearly independent if and only if the matrix having these vectors as its columns has rank n. Proof. Let {v1 , v2 , . . . vn } be the vectors in question, and denote by A the matrix formed by taking these vectors as the columns. Recall that {v1 , v2 , ...
... 1.3, 24 Explain why a set of n vectors in Rn is linearly independent if and only if the matrix having these vectors as its columns has rank n. Proof. Let {v1 , v2 , . . . vn } be the vectors in question, and denote by A the matrix formed by taking these vectors as the columns. Recall that {v1 , v2 , ...
Math 244 Quiz 4, Solutions 1. a) Find a basis T for R 3 that
... The general solution is y = yc + yp = c1 cos(x) + c2 sin(x) + 1. Observe that y 0 = −c1 sin(x) + c2 cos(x). The initial condition y(0) = −1 implies that −1 = c1 + 1 so c1 = −2. The condition that y 0 (0) = −1 implies that −1 = c2 and the solution is y = −2 cos(x) − sin(x) + 1. 3. Obtain the general ...
... The general solution is y = yc + yp = c1 cos(x) + c2 sin(x) + 1. Observe that y 0 = −c1 sin(x) + c2 cos(x). The initial condition y(0) = −1 implies that −1 = c1 + 1 so c1 = −2. The condition that y 0 (0) = −1 implies that −1 = c2 and the solution is y = −2 cos(x) − sin(x) + 1. 3. Obtain the general ...
Basis (linear algebra)
Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.