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I'm studying the book Linear Algebra by Hoffman and Kunze and on page 211 they state the following observation about projections:

Projections can be used to describe direct-sum decompositions of the space V. For, suppose $V=W_1\oplus\ldots \oplus W_k$. For each $j$ we shall define an operator $E_j$ on $V$. Let $\alpha$ be in $V$, say $\alpha = \alpha_1+\ldots+\alpha_k$ with $\alpha_i$ in $W_i$. Define $E_j\alpha=\alpha_j$. Then $E_j$ is a well-defined rule. It is easy to see that $E_j$ is linear, that the range of $E_j$ is $W_j$, and that $E_j^2=E_j$. The null space of $E_j$ is the subspace

$$(W_1+\ldots+W_{j-1}+W_{j+1}+\ldots +W_k)$$

I didn't understand the last line, why $(W_1+\ldots+W_{j-1}+W_{j+1}+\ldots +W_k)$? shouldn't be $$(W_1\oplus\ldots\oplus W_{j-1}\oplus W_{j+1}\oplus\ldots \oplus W_k)$$

EDIT

The user @Semiclassical pointed out a possible typo in the book, but the link provided doesn't have any relationship with the statements of my question. Besides that, the page is also different.

user42912
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  • There's a working list of typos in Hoffman & Kunze in this question: https://math.stackexchange.com/questions/437253/is-there-a-list-of-all-typos-in-hoffman-and-kunze-linear-algebra. In particular, the top-voted answer indicates that there's a typo on page 212: "It should be V=W1⊕⋯⊕Wk." So that may be the issue. – Semiclassical Jan 28 '21 at 18:54
  • @Semiclassical nfortunately not. This typo is about another sentence. – user42912 Jan 28 '21 at 19:01
  • what does the fact that $V$ is the direct sum of $W_i$'s tell you? – user8675309 Jan 28 '21 at 22:06
  • @user8675309 it tells me every vector in $V$ can be represented uniquely as a sum of the $W_i$'s – user42912 Jan 29 '21 at 19:50
  • in other words it tells you that $V= W_1+...+W_k$ and the $W_i$ are independent. Now define $V':=(W_1+\ldots+W_{j-1}+W_{j+1}+\ldots +W_k)$ by independence you also have $V' =(W_1\oplus\ldots\oplus W_{j-1}\oplus W_{j+1}\oplus\ldots \oplus W_k)$ – user8675309 Jan 29 '21 at 22:36
  • @user8675309 that's what I thought, then the book has a typo? – user42912 Jan 31 '21 at 02:22
  • why do you think it is a typo? An equality means the authors may write it either way. – user8675309 Jan 31 '21 at 20:09

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