How does vector algebra work?
Normally
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| Term | Definition |
|---|---|
| How does vector algebra work? | Normally |
| What is the triangle identity for vectors? | |x+y| <= |x| + |y|. |
| What is \hat{x}? | What is \hat{x}? The normalization of x, defined by 1/|x| (x). |
| What is the most important identity in the dot product | |x * y| <= |x||y|. |
| What is the cosine dot product relation | x * y = |x||y|cos(\theta) (x, y \ne 0) |
| Define orthogonal. | x * y = 0 |
| Define Parallel | x * y != 0 x,y != \vec{0}. |
| Defn Vector Equation of Line. | OP + cd. where d is a direction vector. |
| Is x \cross y = y \cross x? | No, x \cross y = (-y \cross x). |
| Define a normal vector to a plane. | For some points P and Q on the plane, a normal vector N is a vector orthogonal to that vector. |
| What is the scalar equation of a plane (in R3)? | n1(x_1) + n2(x_2) + n3(x3) = c. Where n is a normal vector, and c is a constant scalar. |
| What makes two planes parallel? | Their normal vectors are parallel. |
| Define proj and perp. | Given x, y in r. proj_x (y) is the projection of y onto x. Eg. The component of y in the direction of x. perp_x(y) is the component of y orthogonal to x. |
| perp + proj = ? | y. |
| Define REF(A). | (a) All zero rows at bottom. (b) Each leading entry is to the right of all leading entries above it. |
| Define RREF(A) | (a) All zero rows at bottom. (b) Each leading entry is to the right of all leading entries above it. (c) All leading entries are 1. (d) Each leading entry is alone in it's col. |
| Given the SRT, if the system is consistent what is true. | (a) rank(A) = rank(Augmented) (b) #of params = n - rank(A). (c) if rank(a) = n, then the system is consistent for every b in R. |
| What is an underdetermined system of equations. (m equations, n variables). | n > m (more variables than equations). If consistent, will have parameters (infinitely many solutions). |
| What is an overdetermined system of equations. (m equations, n variables). | n < m (more equations that variables). |
| What is the associated homogenous system of [A|b]? | [A|0]. |
| If x is a particular solution to a given system, then: | x + s is a solution given s is a solution to the associated homogenous system. |
| (A^T)_ij | Aji |
| (A^T)^T | A |
| (AB)^T | B^T x A^T |
| Anxm x B_ixj is defined if and only if. | m = i. |
| Inv(cA) | 1/c x Inv(A) |
| Inv(AB) | Inv(B)inv(A) |
| Inv(A^k) | Inv(A)^k for k pos. |
| Matrix Inv Criteria. Mnxn. | (a) rank(A) = n. (b) rref(A) = I. (c) A^T is inv. (d) For all b in rn ax is unique and consistent. (e) Null(A) = {0}. (f) The cols of A are linearly independent. (g) The columns of a form a basis for Rn. |
| Define the span of a set of vectors S. | The set of all linear combinations of those vectors. |
| If Span{S} = B | S spans B. |
| If x is in the col space of [V_1 ... V_k] | x is in the span of V_1 ... Vk. |
| if rank([V_1 ... V_k]) = k. | Then V_1 ... Vk spans R_n. |
| You need at least how many vectors to span Rn. | n. |
| A set of vectors is linearly dependent if | For some vector in the set, that vector is in the span of the other vectors. 0 = c1x_1 + ... + cnx_n where some of the scalars are nonzero. |
| Given K vectors in Rn, S is linearly independent if and only if. | rank(A) = k. |
| Given S and A with cols from S. The linearly independent set S' is made of the vectors that | Have a leading entry in their col in REF. |
| What is a subspace of Rn. | A subset where: (a) The zero vector is included. (b) The subset is closed under vector addition. (c) The subset is closed under scalar mul. |
| For any set of vectors, a span is always: | A subspace of Rn. |
| Each subspace is always expressible | as a linear combination of a set of n vectors. n = dim(U) minimally. |
| B is a basis for U if: | (a) B is linearly independent. (b) U = Span(B). |
| The standard basis for R^n is the set of | N standard basis vectors e_1 through E_n. Corresponding to the n cols of Inxn. |
| Every basis for R^n has | N vectors. |
| B is a basis for R^n if and only if | B is invertable. rank(A) = n. |
| dim(U) equals: | The number of vectors in a B for U. |
| If U is a k > 0 dim subspace of R^n. | (a) a set of more than k vectors in U is linearly dependent. (b) A set of fewer than k vectors cannot span U. (c) A set of k vectors in U spans U if and only if it is linearly independent. |
| How to extend basis? | Put basis and standard basis vectors in matrix. Reduce to REF and use extraction theorem to get an extended basis for R^n containing what you need. |
| What is null(A)? | solution space of Ax = 0. |
| What is col(A) | Span{cols}. (Ax = b defined). |
| rank(A) + nullity(A) = ? for A_mxn? | n. |
| If T: R^n -> R^m [T] is ?,? | mxn |
| n is domain, m is | codomain |
| If det(A) = 0 | Matrix not invertable. |
| + or - first for cofactor expansion. | + |
| det(A) after row/col swap = | -det(A) |
| det(A) after add to other = | det(A) |
| det(A) after col mul by C = | cdet(A) |
| If triangular, determinant = | product along col. |
| det(AB) | det(A)det(B) |
| det(A-1) | 1/det(A) |
| det(A^t) | det(A) |
| To divide (a/w) | mul by (cong(w) / conj(w)). |
| z^-1z = | 1 |
| z + conj(z) z - conj(z) | 2Re(z) 2iIm(z). |
| If w is a root so is | conj(w). |
| (A - \lambda)x = 0 | Eigenspace. |
| If a(lambda) = g(lambda) | A = PDPinv. |
| Reflection of X over Y | 2proj_y(x) - x |