linear combination & linear independence

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$engg2013 lecture 5 linear combination & linear independence * last time how to multiply a matrix and a vector different ways to write down a system of linear equations vector equation matrix-vector product augmented matrix kshum engg2013 * column vectors engg2013 \begin{bmatrix} 2 & 0\\ 1 & -1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 4\\ 5 \end{bmatrix} * review: matrix notation in engg2013, we use capital bold letter for matrix. the first subscript is the row index, the second subscript is the column index. the number in the i-th row and the j-th column is called the (i,j)-entry. cij is the (i,j)-entry in c. kshum engg2013 * m  n engg2013 \mathbf{c} = \begin{bmatrix} c_{11} & c_{12} & c_{13} & \ldots & c_{1n}\\ c_{21} & c_{22} & c_{23} & \ldots & c_{2n}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ c_{m1} & c_{m2} & c_{m3} & \ldots & …$

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$engg2013 * y x 1 1 y x c c c is any real number engg2013 representing a straight line by vector kshum engg2013 * y x c c y x y=x any point on the line y=x can be written as engg2013 adding one more vector kshum engg2013 * y x y=x y x y=x+1 engg2013 \big\{ c\begin{bmatrix} 1\\ 1 \end{bmatrix} + \begin{bmatrix} 0\\ 1 \end{bmatrix}\in \mathbb{r}^2:\, c \text{ is any real number}\big\} * we can add another vector and get the same result kshum engg2013 * y x y=x+1 y x y=x+1 = engg2013 \big\{ c\begin{bmatrix} 1\\ 1 \end{bmatrix} + \begin{bmatrix} 0\\ 1 \end{bmatrix}\in \mathbb{r}^2:\, c \text{ is any real number}\big\} * the whole plane kshum engg2013 * y x scanner engg2013 \big\{ c\begin{bmatrix} 1\\ 1 \end{bmatrix} + \begin{bmatrix} 0\\ 1 \end{bmatrix}\in \mathbb{r}^2:\, c \text{ is any real number}\big\} * question 1 kshum engg2013 * can you find …$

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$atrix} 2\\ 4\\ 6 \end{bmatrix}+ z \begin{bmatrix} 4\\ 8\\ 12 \end{bmatrix}= \begin{bmatrix} 3\\ 6\\ 9 \end{bmatrix} * algebra for linear equations kshum engg2013 * engg2013 review on notation a vector is a list of numbers. the set of all vectors with two components is called . is a short-hand notation for saying that v is a vector with two components the two components in v are real numbers. kshum engg2013 * engg2013 the set of all vectors with three components is called . is a short-hand notation for saying that v is a vector with three components the three components in v are real numbers. kshum engg2013 * engg2013 the set of all vectors with n components is called . we use a zero in boldface, 0, to represent the all-zero vector kshum engg2013 * engg2013 definition: linear combination given vectors v1, v2, …, vi in , and i real …$

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$7 2 4 3 5 is a linear combination of and , because we therefore say that engg2013 mathematical language kshum engg2013 * president obama is not a chinese. ordinary language mathematical language let c be the set of all chinese people. president obama engg2013 example kshum engg2013 * x y z engg2013 x \begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}+ y \begin{bmatrix} 2\\ 4\\ 6 \end{bmatrix}+ z \begin{bmatrix} 4\\ 8\\ 12 \end{bmatrix}= \begin{bmatrix} 3\\ 6\\ 9 \end{bmatrix} * a fundamental fact let a be an mn matrix b be an m1 vector let the columns of a be v1, v2,…, vn. the followings are logically equivalent: kshum engg2013 * we can find a vector x such that 1 2 3 “logically equivalent” means if one of them is true, then all of them is true if one of them is false, then all of them is false. engg2013 theorem 1 with …$

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$re in the examples with infinitely many solutions the common feature is that one of the vector is a linear combination of the others. notice that is a scalar multiple of is a linear combination of and engg2013 \begin{bmatrix} 1 & 2 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}= \begin{bmatrix} 5\\ 7\end{bmatrix} \text{ is not solvable.} * definition: linear dependence vectors v1, v2, …, vr are said to be linear dependent if we can find r real number c1, c2, …, cr, not all of them equal to zero, such that 0 = c1 v1+ c2 v2+ …+ cr vr otherwise, are v1, v2, …, vr are said to be linear independent. in other words, v1, v2, …, vr are be linear independent if, the only choice of c1, c2, …, cr, such that 0 = c1 v1+ c2 v2+ …+ cr vr is c1 = c2 = …$

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engg2013 lecture 5 linear combination & linear independence * last time how to multiply a matrix and a vector different ways to write down a system of linear equations vector equation matrix-vector product augmented matrix kshum engg2013 * column vectors engg2013 \begin{bmatrix} 2 & 0\\ 1 & -1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 4\\ 5 \end{bmatrix} * review: matrix notation in engg2013, we use capital bold letter for matrix. the first subscript is the row index, the second subscript is the column index. the number in the i-th row and the j-th column is called the (i,j)-entry. cij is the (i,j)-entry in c. kshum engg2013 * m  n engg2013 \mathbf{c} = \begin{bmatrix} c_{11} & c_{12} & c_{13} …

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