# Questions tagged [linear-algebra]

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

80,455 questions
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### Applying a Sigma-Point Kalman Filter to State of Charge Estimation

I've found a research paper that would allow me to implement a Kalman filter to estimate state of charge for a LiPo battery, but cannot make sense of some of the symbols as I'm a first year. Can ...
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### Elementary transformations of the set of vectors with respect to which the linear span is invariant

Let $L$ denote a linear space. Let $x_1,x_2,\cdots,x_n \in L$. Observe that $span\{x_1,x_2,\cdots,x_n\} = span\{x_2,x_1,\cdots,x_n\}$ This motivates the following question: under what operations ...
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### How to determine the range of unknown values ​to make the matrix positive semi-determined?

$$\left[ \begin{matrix} a & -1 & -2 & -2 \\ -1 & b & -2 & -2 \\ -2 & -2 & c & -1 \\ -2 & -2 & -1 & d \end{matrix} \right] \tag{1}$$ ...
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### Let $A$ be $2 \times 2$ nonzero real matrix.which of the following is true?

Let $A$ be $2 \times 2$ nonzero real matrix.which of the following is true? $(A)$ trace of $A^2$ is positive $(B)$ $A$ has non zero eigenvalue. $(C)$ All entries of $A^2$ can't be ...
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### Show that this set is a basis for $S_1+S_2$

Let $$B_1 = \{v_1, \dots v_n, x_1 \dots x_r\}$$ $$B_2=\{v_1 \dots v_n, y_1 \dots y_s\}$$ $$B_3 = \{v_1 \dots v_n\}$$ be basis for subspaces $S_1$ , $S_2$ and $S_1 \cap S_2$ respectively. Show that ...
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### Can $\| \left( X'X + \lambda I \right) ^{-1} X'y \| = t$ be solved for $\lambda$?

In this post I suggested that the expression $$\| \left( X'X + \lambda I \right) ^{-1} X'y \| = t$$ couldn't be easily solved for $\lambda$, because you need to "invert" the norm. But in general ...
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### Determine the values of a, b for which the systems have (1) exactly one solution, (2) no solutions, (3) infinitely many solutions.

I'll leave two pictures, can someone check if I'm right? (exercise b)
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### What kind of matrices can be visualized by graphing the column vectors?

In the 3Blue1Brown video “Linear transformations and matrices” linear transformations are visualized by overlaying gridlines which have a position determined by the values of the transformed basis ...
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### Determine the dimension and find a basis of a vector space

$V_1 = (x_1, ..., x_7)^T \in \mathbb{Z}^7_{5} : x_1 + 3x_2 +x_3+2x_4+3x_5+x_6+2x_7 = 0$ $3x_1 + 4x_2 +3x_3+x_4+4x_5+2x_6+4x_7 = 0$ $2x_1 + x_2 +4x_3+4x_5+x_6+2x_7 = 0$ I am supposed to find ...
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### Determine the values of a, b and c, for which the systems have (1) exactly one solution, (2) no solutions, (3) infinitely many solutions.

I will just attach a picture. Can someone help me to solve this? I think I missed some information.
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### Matrix multiplication, linear transformations and systems of equations

In linear algebra, I learned that the reason why matrix multiplication is defined in its way is because of composition of linear transformations. So, multiplication of two matrices refers to the ...
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