Linear Algebra Chapter 1 Flashcards

Describes a first order line in R^n equations of the form a1x1 + a2x2 + ... + anxn = C, where a1...an are R and x1...xn are variables.

A collection of 1 or more linear equations to be solved simultaneously.

A set of real numbers s1...sn such that each equation holds if we set x1 = s1, ..., xn = sn.

<p>A set of all solutions in a system.</p> Signup and view all the answers

<p>At least one solution exists.</p> Signup and view all the answers

<p>No solutions exist.</p> Signup and view all the answers

<p>Two systems that have the same solution set.</p> Signup and view all the answers

<p>Use interchanging, scaling, and replacement to isolate the variable and then plug into other variables.</p> Signup and view all the answers

<p>Reorder equations.</p> Signup and view all the answers

<p>Multiply each equation by a constant.</p> Signup and view all the answers

<p>Replacing one equation with itself and a multiple of another.</p> Signup and view all the answers

<p>A rectangular array of numbers with m rows and n columns.</p> Signup and view all the answers

<p>A matrix that includes the set of constant solutions (numbers without variables).</p> Signup and view all the answers

<p>Try to make each pivot equal to one, with each pivot being the only number in its column.</p> Signup and view all the answers

<ol> <li>All 0 rows at the bottom of the matrix. 2. Each leading nonzero entry in a row to the right of the leading nonzero entry in the previous row. 3. All entries in the column beneath a leading nonzero entry are 0's.</li> </ol> Signup and view all the answers

<p>All the requirements for Echelon Form plus that each leading nonzero entry is 1 and is the only nonzero entry in its column.</p> Signup and view all the answers

<p>Leading nonzero entry in an echelon matrix.</p> Signup and view all the answers

<p>Position of the pivot in an echelon matrix.</p> Signup and view all the answers

<p>A variable corresponding to a column with a pivot.</p> Signup and view all the answers

<p>A variable corresponding to a column with no pivot; it can be any number.</p> Signup and view all the answers

<p>The echelon form of the matrix has no row of the form [0...0 C] where C is not equal to 0.</p> Signup and view all the answers

<p>The solution is unique and there are no free variables.</p> Signup and view all the answers

<p>Quantities with direction and magnitude.</p> Signup and view all the answers

<p>Quantity with only magnitude.</p> Signup and view all the answers

<p>Multiplication of scalar and vector; the answer is a vector, same direction if scalar is positive, opposite if scalar is negative.</p> Signup and view all the answers

<p>Given v1...vk, the sum vector y = c1v1 + ... + ck vk.</p> Signup and view all the answers

<p>'Weights' of linear combination.</p> Signup and view all the answers

<p>Given a collection of vectors {vi...vk} in Rn, the collection of vectors in Rn which can be expressed as y = c1v1 + ... + ck vk.</p> Signup and view all the answers

<p>Let A be an m x n matrix with columns a1...an. Let x be a vector which is the linear combination of columns a, with weights Ax = x1a1 + x2a2 + ... + anxn.</p> Signup and view all the answers

<p>Ax = b where A is the m x n coefficient matrix, x is a variable vector, and b is the vector of constant terms.</p> Signup and view all the answers

<ol> <li>For all real vectors b, Ax = b is solvable. 2. Every real b is a linear combination of columns of A. 3. The columns of A span R^m. 4. Every row of A has a pivot.</li> </ol> Signup and view all the answers

<p>Pivot in every column means that all variables are basic.</p> Signup and view all the answers

<p>Ax = 0 where 0 is the zero vector; always solvable for any matrix A, just make x = 0.</p> Signup and view all the answers

<p>A solution to a system that makes the system equal to the zero vector, but is not the zero vector itself.</p> Signup and view all the answers

<p>A solution that can only equal 0 with the 0 solution.</p> Signup and view all the answers

<p>A collection of vectors {v1..vk} in some Rn whose linear combination has only the trivial solution.</p> Signup and view all the answers

<p>A collection of vectors {v1..vk} in some Rn whose linear combination has more than just the trivial solution.</p> Signup and view all the answers

<ol> <li>The 0 vector is dependent. 2. If the 0 vector is an element of {v1...vk}, dependent. 3. If vectors are multiples, dependent. 4. If {v1..vk} in Rn and k &gt; n, dependent.</li> </ol> Signup and view all the answers

<p>T: Rn -&gt; Rm is a function taking vectors in Rn to vectors in Rm.</p> Signup and view all the answers

<p>T: Rn -&gt; Rm is Rn.</p> Signup and view all the answers

<p>T: Rn -&gt; Rm is Rm.</p> Signup and view all the answers

<p>Subset of vectors in Rm which are T(x) for some vector x in Rn.</p> Signup and view all the answers

<ol> <li>T(u + v) = T(u) + T(v). 2. T(cu) = cT(u).</li> </ol> Signup and view all the answers

<p>A T: Rn -&gt; Rm whose image is all of Rm; recall b in Rm so that there is x in Rn with T(x) = b.</p> Signup and view all the answers

<p>A T: Rn -&gt; Rm which if T(x1) = T(x2) implies x1 = x2.</p> Signup and view all the answers

<p>Let A be an m x n matrix and B be an n x k matrix. Then the product is given by AB = matrix Ab1 where b1 is a column of B.</p> Signup and view all the answers

<ol> <li>A(BC) = AB(C). 2. A(B + C) = AB + AC. 3. r(AB) = (rA)B = A(rB). 4. If A (mxn) and ImA = A = AIn.</li> </ol> Signup and view all the answers

<p>Multiply by itself; can only happen with a square matrix.</p> Signup and view all the answers

<p>Flips matrix; rows become columns and columns become rows, denoted At.</p> Signup and view all the answers

<ol> <li>(At)t = A. 2. (A + B)t = At + Bt. 3. (rA)t = r(At). 4. (AB)t = BtAt; must swap order!</li> </ol> Signup and view all the answers

<p>A is invertible if there is some nxn matrix B such that AB = In = BA; we denote B by A^-1.</p> Signup and view all the answers

<p>A matrix that is not invertible.</p> Signup and view all the answers

<p>If A = [a b; c d] is a 2x2 matrix then A is invertible iff ad - bc is not equal to 0, then A^-1 = 1/(ad - bc) [d -b; -c a].</p> Signup and view all the answers

<ol> <li>ax = b has one solution for every b in Rn. 2. Every matrix has n pivot positions. 3. If A is invertible, so is A^-1. 4. (AB)^-1 = B^-1A^-1. 5. If A is invertible, then so is At.</li> </ol> Signup and view all the answers

<p>[A In] -&gt; [In A]; the last three columns are the inverse.</p> Signup and view all the answers

<p>A matrix that is obtained from the identity matrix with a single row operation.</p> Signup and view all the answers

<p>Of [a] is a; A = a11detA11 - a12detA12 + ... + (-1)^n a1n detA1n.</p> Signup and view all the answers

<p>Square matrices.</p> Signup and view all the answers

<p>The matrix has an inverse.</p> Signup and view all the answers

<p>The determinant is 0.</p> Signup and view all the answers

<p>The ijth cofactor of A is (-1)^(i+j) Aij.</p> Signup and view all the answers

<p>The process of finding a determinant.</p> Signup and view all the answers

<p>|x1 # # | 0 |x2 # | 0 0 x3 ...</p> Signup and view all the answers

<p>|x1 0 0 | # x2 0 | # # x3 ...</p> Signup and view all the answers

<p>Multiply the diagonal leading entries.</p> Signup and view all the answers

<p>Replacement = nothing; exchange = multiply by -1; scaled by k = multiply by k.</p> Signup and view all the answers

<p>det A = (-1)^r * product of pivots, where r is the number of pivots.</p> Signup and view all the answers

<ol> <li>det At = det A. 2. det(AB) = detA * detB for A and B that are nxn matrices. 3. det(A^-1) = 1/detA.</li> </ol> Signup and view all the answers