If \(\mathrm{T}:\mathbf{R}^{n}\to\mathbf{R}^{m}\) is a linear map, then it is represented by a matrix of order :
\(m\times n\)
The matrix of \(\mathrm{T}\) relative to the standard bases has as its \(j\)-th column the coordinate vector of \(\mathrm{T}(e_{j})\), for \(j=1,\ldots,n\).
Each \(\mathrm{T}(e_{j})\) lies in \(\mathbf{R}^{m}\), so every column has \(m\) entries, and there are \(n\) columns. Then \(\mathrm{T}(x)=Ax\) with \(x\in\mathbf{R}^{n}\) a column of length \(n\) requires exactly \(n\) columns in \(A\).
So \(A\) has \(m\) rows and \(n\) columns. Square shapes \(m\times m\) or \(n\times n\) only fit when \(m=n\), and \(n\times m\) would act on vectors of length \(m\), not on \(\mathbf{R}^{n}\).
Check: \(\mathrm{T}(x,y,z)=(x+y,\ z)\) maps \(\mathbf{R}^{3}\) to \(\mathbf{R}^{2}\) and has matrix \(\begin{bmatrix}1&1&0\\0&0&1\end{bmatrix}\), which is \(2\times 3\), that is \(m\times n\).
Hence, \(m\times n\).
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\(\frac{2}{1} + \frac{3}{4} + \frac{4}{9} + \ldots + \frac{n+1}{n^2} + \ldots\)
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.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
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(i) Kernel
(ii) Interface
(iii) Nucleus
(iv) Compiler
Codes:
Which of the following are referred to as low level language?
(i) Machine language
(ii) Assembly language
(iii) High level language
(iv) Natural language
Codes:
Match the following:
List I List II (a) New York Public Library (i) Shelv AR (b) Miami University Augmented Research Group (ii) Smart library (c) Oula University Library (iii) Wolf Walk (d) North Carolina State University (iv) Find the future Codes:
Which part of the QR code represents the encoded data?
