Flashcards
GRE Study Linear Algebra
Cards in this deck (26)
- 1Number of Solutions to a Linear System0 solutions (inconsistent)
1 solution (consistent)
infinitely many solutions (consistent) - 2Dot Product\( \vec{u}\cdot\vec{v} = u_{1}v_{1} + u_{2}v_{2} + \ldots + u_{n}v_{n}\)
- 3Socks and Shoes Theorem\((AB)^{-1} = A^{-1}B^{-1} \)
- 4Number of free variables =number of unknowns - number of nonzero rows in echelon matrix
- 5Solution to \(A\vec{x} = \vec{b} \) if \(A\) is invertible\(\vec{x} = A^{-1}\vec{b} \)
- 6Gaussian Elimination1) augmented matrix
2) reduce to echelon form
3) backwards sub - 7Reducing to Echelon form options> multiply row by constant
> interchange two rows
> add multiple of another row - 8Inverse of a 2 by 2 Matrix \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]\[ A^{-1} = \frac{1}{ad-bc}
\begin{bmatrix}
d & -b \\
-c & a
\end{bmatrix} \] - 9Vector Space> closed under addition and scalar multiplication
> must contain \(\vec{0} \) - 10Nullspace> set of all solutions to \(A\vec{x} = \vec{0} \)
> if a matrix is invertible, the only solution to \(A\vec{x} = \vec{0} \) is the trivial one - 11Linear Combination\(k_{1}\vec{v_{1}} + k_{2}\vec{v_{2}}+ \ldots k_{n}\vec{v_{n}}\)
- 12Spanset of all linear combinations of vectors
- 13Linearly IndependentIf \(k_{1}\vec{v_{1}} + k_{2}\vec{v_{2}}+ \ldots k_{n}\vec{v_{n}} = \vec{0}\) is only true for \(k_{i} = 0\)
- 14Basiscollection of linearly independent vectors that span a space
- 15Dimensionnumber of vectors in a space
- 16To determine of vectors are linearly independent......echelon the vectors; any free variables means they are dependent
- 17Cross Product\(\vec{u}\times\vec{v} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
u_{1} & u_{2} & u_{3} \\
v_{1} & v_{2} & v_{3}
\end{vmatrix} \) - 18Equation for plane through two vectors> cross product
> use cross product to get a normal vector \( (a b c) \)
> plane: \(ax +by +cz = 0\) - 19Column Rankmax number of linearly independent columns
- 20Row Rankmax number of linearly independent rows
- 21Rank =Row rank = column rank
- 22Column Spacefor an m by n matrix, CS is a subspace of \( R^{m}\) spanning the columns
- 23dim(CS(A)) =rank(A)
- 24Basis for CS> echelon form of \(A^{T}\)
> find number of independent columns
> pick those columns - 25To see if \(\vec{b}\) is in CS(A)......see if there is a solution to \(A\vec{x} = \vec{b}\)
- 26—
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- Flashcards on GRE Study Linear Algebra, created by Marissa Miller on 05/10/2015.
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- Marissa Miller
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