In this note,i will make some summaries about vectors
Vectors and Linear Combination
We often write the vector as (1,2) for example,but in linear algebra,we try to define it as \(\begin{bmatrix} 1\\ 2\end{bmatrix}\)
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A vector v in two-dimensional space has two components \(v_1\) and \(v_2\)
,when in three-dimensional will have three components -
the $c v +d w $ is the linear combination of the vectors \(v\) and \(w\) (\(c,d\in R\))
A linear combination of three vectors u and v and w is cu+dv+ew -
Vectors obey the parallelogram law
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The linear combinations of vectors can span a space
Length and Dot Products
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The dot product \(v·w\) multiplies each component \(v_i\) by \(w_i\) and adds all \(v_iw_i\)
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The length \(|v|\) is the square root of \(v·v\) .Then \(u=\frac{v}{|v|}\) is unit vector:length 1.
- for example:The length of\(v=\begin{bmatrix} v_1\\ v_2\\..\\v_n\end{bmatrix}\)is \(\displaystyle \sqrt(\sum_{i=1}^{n}(v_i)^2)\)
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The dot product is \(v·w=0\) when vectors v and w are perpendicular.
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The cosine of \(\theta\)(the angle between any nonzero v and w) never exceeds 1:
- Cosine \(\cos\theta=\frac{v·w}{|v||w|}\)
- \(|v·w|<=|v||w|\)
Matrices
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Matrix: A matrix is a higher-dimensional structure composed of multiple vectors. A ( m \(\times\) n ) matrix can be viewed as consisting of ( m ) column vectors or ( n ) row vectors.
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Matrix times vector : Ax=combination of the columns of A.
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The solution to \(Ax=b\) is x=\(A^{-1}b\),when A is an invertible matrix.
- an invertible matrix is typically written as \(A^{-1}\).When $AA^{-1} = I $ holds, it indicates that the inverse \(A^{-1}\) exists.
- In this context, \(I\) represents the identity matrix
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A singular matrix is a square matrix that does not have an inverse.because Its columns lie in the same plane